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X 


AN 
ELEMENTAET    TEEATISE 

ON   THE 

?  •  ■    '  \r^ 

THEORY  OF  DETERMINANTS. 

A    TEXT-BOOK  FOR  COLLEGES. 


PAUL   H.   HANUS, 

M 
Formerly  Professor  of  Mathematigs  in  the  University  of  Colorado; 
NOW  Principal  of  Denver  High  School,  District 
No.  2  (West  Denver). 


UHV 


BOSTON: 
PUBLISHED  BY  GINN  AND  COMPANY. 

1886. 


Entered,  according  to  the  Act  of  Congress,  in  the  year  M86,  by 

PAUL   H:   IIAXUS, 

in  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


J.  8.  Gushing  &  Co.,  Pkintkrs,  Boston. 


PEEFAOE. 


The  importance  of  a  knowledge  of  Determinants  to  all 
who  extend  their  reading  beyond  the  elements  of  mathematics, 
and  the  fact  that  most  modern  writers  employ  the  determinant 
notation,  have  led  to  the  belief  that  an  American  work  on 
Determinants  might  satisfy  a  growing  demand. 

This  is  a  text-book,  and  not  an  exhaustive  treatise.  Enough 
is  given,  however,  to  enable  the  student  to  use  the  determinant 
notation  with  ease,  and  to  enable  him  to  pursue  his  further 
reading  in  the  modern  higher  mathematics  with  pleasure  and 
profit. 

The  book  is  written  with  reference  to  the  wants  of  the 
private  student  as  well  as  to  the  needs  of  the  class-room.  The 
subject  is  at  first  presented  with  great  simplicity.  As  the  stu- 
dent advances,  less  attention  is  given  to  details.  More  than 
half  the  volume  is  devoted  to  applications  and  special  forms, 
that  the  reader  may  get  some  notion  of  the  power  and  utility 
of  determinants  as  instruments  of  research. 

Throughout  the  work  care  has  been  taken  to  show  how  each 
new  concept  has  been  evolved  naturally ;  and,  whenever  it  is 
thought  advisable,  a  special  case  precedes  the  general  dis- 
cussion. 

The  work  has  been  written  in  the  far  West,  where  contact 
with  others  in  the  same  field   was   practically  impossible.      I 


167219 


iv  PREFACE. 

shall  therefore  be  grateful  for  any  notification  of  errors  that 
may  have  escaped  detection. 

My  thanks  are  due  to  Messrs.  J.  S.  Gushing  &  Co.,  of 
Boston,  for  great  care  and  patience  manifested  in  the  prepara- 
tion of  the  plates. 

Among  the  works  consulted  most  assistance  has  been  derived 
from  the  following.  All  the  works  named  have  been  used 
freely. 

Matzka.  —  Grundziige  der  systematischen  Einfiihrung  und  Begriin- 
dung  der  Lehre  der  Determinanten. 

Baltzer.  —  Theorie  und  Anwendung  der  Determinanten  (Fiinfte 
Aufiage). 

Gunther.  —  Lehrbuch  der  Deterniinanten-Theorie  (Zweite  Aufiage). 

Diekmann.  —  Einleitung  in  die  Lehre  von  den  Determinanten  und 
ihrer  Anwendung  auf,  etc. 

Dostor.  —  Elements  de  la  Theorie  des  Determinants  avec  Applica- 
tions, etc.  (Deuxieme  edition). 

Houel. —  Cours  de  Calcul  Infinitesimal. 

Scott.  —  A  Treatise  on  the  Theory  of  Determinants  and  their  Appli- 
cations, etc. 

Burnside  and  Panton.  —  The  Theory  of  Equations,  with  an  Intro- 
duction, etc. 

Muir.  —  A  Treatise  on  the  Theory  of  Determinants. 

I  am  especially  indebted  to  the  last,  two  works  for  many 
examples. 

PAUL   H.   HAN  US. 

BouLDBU,  Col.,  May,  1886. 


co]:^te:n"ts. 


CHAPTER  I. 
Preliminary  Xotioxs  and  Definitions. 

AET.  PAGE. 

1.     Discovery  of  Determinants 1 

2-7.     Determinants  produced  by  eliminating  the  unknowns 

from  a  system  of  simultaneous  equations         .         .  2-8 

8-10.     Values  of  the  unknowns  in  determinant  form     .         .  8-10 

9.     Change  of  sign 9 

11.     iiTotation 11-12 

12-14.     Expansions  with  square  notation          ....  13 

15.     Rule  for  expanding  a  determinant  of  the  third  order  .  14 

Examples '  14-16 

CHAPTER  n. 
General  Properties  of  Determinants. 

16-19.     Definition  and  notation         .         .         .         .        .         .  17-20 

20.  Corollaries 20 

21.  Inversions  of  order 20-21 

22.  Number  of  terms  in  a  determinant      .        .         .        .  21 
23-24.     Corollaries;  expansions 22-23 

25.  If  the  rows  in  order  are  made  the  columns  in  order,  etc.       23 

26.  Number  of  positive  and  negative  terms        ...  24 

27.  Interchange  of  two  parallel  lines 24 


VI  "  CONTENTS. 

AUT.  PAGE. 

28.  Two  identical  parallel  lines 25 

29.  Cyclical  permutations   .     r    • 25-26 

30.  Corollary 26 

Examples 27 

31.  Every   element   of   a   line    multiplied    by  the   same 

number 27 

32-33.     Corollaries 28 

34-35.     Decomposition   of  determinant  with   polynomial  ele- 
ments          28-29 

36.  Converse  of  34       .         .         .         .    '    .        .         .         .  30 

37.  Transformation  by  addition  of  parallel  lines       .         .  30-31 

38.  Minor  determinants,  or  Minors    .         .         ...         .  31-32 

39.  Expansion  of  determinant  as  linear  function  of  the 
elements  of  one  line .         .         .         .         .         .         .  32 

40.  Coefficient   of   any   element   in   the   expansion    of    a 
determinant 33-34 

41-44.     Corollaries ;  expansions 34-36 

Examples 3G-40 

45.     Elements  of  a  line  multiplied  by  first  minors  of  corre- 
sponding elements  of  a  parallel  line         .         .         .  10-41 
47.     Expansion  in  zero-axial  determinants  ....  41-42 
48-49.     Simplification  by  taking  each  consecutive  pair  of  ele- 
ments in  the  first  row,  etc 43-44 

Examples       .         .         .         .         ...         .         .         .  45-46 

50-54.     The  product  of  two  determinants         ....  40-53  " 

Examples 53-56 

55.     Laplace's  Theorem  (expansion) 56-57 

57.  Product  of  a  determinant  by  one  of  its  minors     .         .  58-60 

58.  Rectangular  Arrays  or  Matrices  (product  of)      .         .  61-63* 
59-62.     The  Reciprocal  or  adjugate  determinant      .         .         .  63-68 

Examples 69-71 

63.     Special    expansions    (including    Cauchy's    Theorem, 

Example  III.)          .        .        .        .        .        .        .  71-76 


CONTENTS. 


Yll 


64.     Solutions  of  certain  determinant  equations  . 
65-66.     Differential  of  a  determinant 


PAGE. 

76-78 
79-81 


CHAPTER  III. 
Applications  and  Special  Forms. 

67-70.     Solution    of    a   system    of   simple   simultaneous 

equations 82-86 

71-72.     If  the  equations  of  the  system  are  not  indepen- 
dent    86 

73.     If  mi  =  m2=  '•'  =mn-i  =  0,  and  one  m  as  wi«  does 

not 87 

74-78.     Solution   of    a  system  of    linear    homogeneous 

equations 87-92 

1  i     77.     The    condition  A  =  0    for    a    system    of    homo- 
geneous equations 91 

79.  Condition  fulfilled  when  n  equations  containing 

n— 1  unknowns  are  simultaneous    .        .         .  92-93 

80.  Solution  of  a  system  of  simple  equations  by  79     .  93-94 

81.  Another  application  of  79 94 

82-83.     The  Matrix  as  the  result  of  elimination        .        .  95-97 

84-89.     Applications  of  preceding  processes      .        .        .  98-110 

90-100.     Resultants,  or  Eliminants 110-126 

91.  Euler's  ]\Iethod  of  elimination       ....  111-112 

92.  Sylvester's  Method         .        .        .        .        .        .  113-114 

93.  Bezout's  Method  (Cauchy) 114-118 

94-95.  Resultant  in  terms  of  the  roots      ....  118-121 

96.     Properties  of  Resultant         .        .        .        .  '     .  121-122 

97-100.     Applications  of  Sylvester's  Method       .        .        .  122-126 

101-102.     Discriminant  of  an  equation  ....  126-129 

103.     Resultant  of  a  system  of  homogeneous  equations 

when  n  — 1  are  linear,  and  one  is  of  the  second 

degree        129-130 


VIU 


CONTENTS. 


ART.  PAGE. 

104—105.     Special  solutions  of  simultaneous  quadratics        .  131-136 

106.     Solution  of  the  Cubic     .        .        ....  137-138 

107-112.     Symmetrical  determinants,  definitions,  and  some 

special  properties 138-144 

113.     If  X  be  subtracted  from  each  element  of  the  prin- 
cipal diagonal  of  a  symmetrical  determinant    .  144-146 
114-118.     Orthosymmetric  determinants       .         .        .  •     .  146-150 
119-126.     Skew  determinants  and  skew  symmetrical  deter- 
minants        151-159 

127-131.     Pfaffians 159-164 

132-134.     Circi:^lants 164-169 

135.     Centrosymmetric  determinants      ....  169-171 

136-150.     Continuants 171-186 

151-163.     Alternants 187-201 

164-170.     TheJacobian          .......  201-209 

171.     The  Hessian 209 

172-176.     The  Wronskian 209-212 

177-179.  •  Linear  substitution ;  Orthogonal  substitution       .  213-217 


THEORY   OF   DETERMINANTS. 


CHAPTER  I. 

'    PEELIMINARY  NOTIONS   AND   DEFINITIONS. 

1.  The  first  notion  of  Determinants  we  owe  to  Leibnitz,  who, 
in  his  attempts  to  simplify  the  expressions  arising  in  the  elimi- 
.nation  of  the  unknown  quantities  from  a  set  of  linear  equations, 
employed  symbols  nearlj'  identical  with  our  present  determinant 
notation.  In  a  letter  dated  April  28,  1693,  Leibnitz  communi- 
cates his  discovery  to  LTIospital ;  and  later,  in  another  letter, 
expresses  the  conviction  that  the  functions  will  develop  remark- 
able and  very  important  properties,  —  a  conviction  which  time- 
has  abundantly  verified.  Leibnitz,  however,  never  pursued  the 
subject  himself,  and  his  discovery  lay  dormant  till  the  middle 
of  the  eighteenth  century. 

In  1750  the  celebrated  geometer,  Gabriel  Cramer,  rediscovered 
determinants  while  working  upon  the  analysis  of  curves.  Dur- 
ing the  course  of  his  investigations,  Cramer  had  to  solve  sets 
of  linear  equations,  and  naturally  encountered  the  same  func- 
tions that  had  attracted  the  attention  of  Leibnitz.*  To  Cramer 
is  due  the  general  rule  for  the  solution  of  n  simultaneous  linear 
equations  (non-homogeneous),  containing  as  many  unknown 
quantities. 

This  rule  was  inferred  without  proof  from  the  form  of  the 
values  of  the  unknown  quantities  obtained  in  solving  sets  of 
two  and  three  equations. 


♦  The  particular  problem  which  led  to  Cramer's  discovery  of  deter- 
minants appears  to  have  been  :  To  pass  a  curve  of  the  uth  order  through 

u'^      8y     .  .    . 

any  — | given  pomts. 

2,       2, 


2  THEORY   OF   DETERMINANTS. 

Since  the  time  of  Cramer  important  advances  hare  been 
made.  The  names  of  many  celebrated  mathematicians  appear 
in  the  list  of  those  who  aided  the  evolution  of  a  theory  of  deter- 
minants. Prominent  among  these  are  Vandermonde  and  Gauss. 
From  Gauss  the  name  "determinant"  instead  of  "resultant" 
was  adopted  by  Cauchy.  Cauchy  and  Jacobi  are  perhaps  to 
be  considered  as  the  greatest  among  those  who  first  developed 
the  subject.  The  monograph  of  Jacobi,  published  in  1841,* 
established  the  foundation  of  a  treatise  on  the  theory  of 
determinants ;  and  his  own  writings,  as  well  as  the  works  of 
many  eminent  mathematicians  during  the  past  fifty  years,  attest 
the  wonderful  power  of  determinants  as  instruments  of  mathe- 
matical investigation,  and  the  fruitfulness  of  the  functions 
themselves. 

2.  The  most  natural  way  of  approaching  the  theory  of  deter- 
minants would  be  along  the  line  of  development.  This  is 
accordingly  our  purpose.  Owing  to  peculiar  difficulties  attend- 
ing this  mode  of  procedure,  we  can  however  onl}'  employ  this 
.method  at  the  outset,  and  must  soon  adopt  a  presentation 
better  suited  to  the  further  unfolding  of  the  subject,  and  free 
from  the  peculiar  difficulties  alluded  to. 

Determinants  of  the  second^  thirds  and  fourth  order. 

3.  Consider  the  set  of  four  simultaneous  linear  equations  :  — 

(1)  a^x -{- h-^y -\- CiZ -\- d^t  —  rrii 

(2)  a<iX  +  h^y  +  c^z  -j-dzt  =  n^ 

(3)  a;iX-^b.y-\-CsZ-\-dst==ms 

(4)  a^x -\- b^y -^  c^z -{•  d^t  =  m^ 

Here  it  will  be  convenient  to  eliminate  the  unknown  quantities 
in  a  uniform  manner,  as  follows  :  in  each  set  of  equations  to  be 
obtained,  (2)  will  be  multiplied  by  the  coefficient  of  the  un- 
known in  (1)  that  is  to  be  eliminated,  and  (1)  by  the  corre- 
sponding  coefficient  in   (2)  ;    (3)   will   be   multiplied   by  the 

*  De  Formatione  et  Proprietatibus  Determinantium. 


I. 


PRELIMINARY   NOTIONS    AND  J3EF1NITI0NS.  6 

coefficient  of  tlie  unl^nown  under  consideration  in  (2) ,  and  (2) 
by  the  corresponding  coefficient  in  (3)  ;  and  so  on  tlirough  the 
set.  Having  tlius  made  the  coefficients  of  one  of  the  unknowns, 
it',  say,  the  same  in  all  the  equations,  we  will  then  eliminate  x 
by  subtracting  (1)  from  (2),  (2)  from  (3),  etc.  We  shall  find 
in  performing  these  operations  that  the  coefficients  Of  the  un- 
known quantities  and  the  absolute  terra  after  each  elimination 
are  functions  of  a  particular  form,  and  subject  to  the  same  law 
of  formation, — that  these  functions  are,  in  fact,  Determinants. 
Eliminating  x  in  set  I.  as  directed,  we  have 

(1)  (di^s  — <^2^l)2/+(^1^2  — «2Cl)^+(<^l<^2— %^l)^=<^l*^2  — «2^ll 

(2)  {a,^z—aJ)<^y-\-  (r/oCs— agCs)^;^  {a2dz—a.jic^t=a.2m^—a^m2  \  II. 

(3 )  ( a^^ — a^dg)  y  +  (^304 — a^c^  z  +  {a-^d^ — a^d.^  t = a^m^^ — a^m^  j 

4.  Examining  these  binomial  coefficients,  we  see  that  each 
contains  one  positive  and  one  negative  term,  and  involves  four 
quantities,  viz.,  ai,  a2,  61,  &2  i  or  ag?  %?  <^2?  Cg,  etc.  It  will  also 
be  noticed  that  each  term  never  contains  more  than  one  a 
(coefficient  of  a?) ,  or  h  (coefficient  of  2/) ,  or  c  (coefficient  of  z) , 
etc.,  but  that  each  term  does  contain  all  the  subscripts  that 
occur  in  the  binomial.  Finally,  the  terms  in  which  the  sub- 
scripts occur  in  their  natural  order  are  positive,  while  in  the 
negative  terms  there  is  an  inversion  of  the  natural  order  in  the 
subscripts,  ^.e.,  a^^c^  is  +,  but  a4C3  is  — .  Such  binomials  are 
determinants  of  the  second  order.*  (The  order  of  a  determinant 
is  determined  b}'  the  number  of  factors  in  each  term.)  It  has 
been  agreed  to  denote  them,  following  Laplace,  by  writing  the 
letters  involved  in  regular  succession,  affecting  each  with  the 
subscripts  in  order,  and  enclosing  the  whole  expression  within 
parentheses,  thus  :  {a^h^)  =  ai^a  —  ^3^1  '•>  {(^2^^)  =  «2^3  —  <^3C2,  etc. 

Introducing  this  notation,  set  II.  becomes 

( 1 )  («! 62)  y  4-  (or-i C2)  z  +  (a-i d^)  t=  {a^m^)  ' 

(2)  {a^h)  y  +  (agCg)  z  +  (ciodg)  t=  {cum^)    .  III. 

(3)  {a^h,)  y  +  (otgC^)  z  +  {a^d^)  t  =  (^3^714) 


*  The  general  definition  of  a  determinant  is  given  in  17,  Chap.  II. 


IV. 


4  THEORY    OF   DETEllMINANTS. 

5.  If  we  now  eliminate  y,  according  to  the  directions  given 
in  3,  we  have 

(1)  [("1&2)    («2C3)-(«2W    (%C,)]2+[(aiW<«2«^3) 

—  (agftg)   (aiC^2)]^=  («1&2)   (^2^3)   —   {^M   («l'^2) 

(2)  t  (ag^s)    («3C4)  -  («3&4)    («2C3)]  2  +  [(a2&3)    («3^4) 

—  (a364)  {a2dz)~\t  =  {a^h)  {a^m^)—  {a^h^)  {a^m^) 

Examining  the  binomial  coefficients  of  the  unknowns,  and 
the  absolute  terms  in  set  IV,  we  see  at  once  that  they  are  of 
the  same  form;  and  if  we  can  simplify  any  one  of  them  and 
discover  the  law  of  formation,  we  have  them  all.  For  this 
purpose  let  us  expand  the  coefficient  of  2;,  putting,  for  short- 
ness, this  coefficient  equal  to  C.  Then,  by  the  definition  in  4, 
G  =  (ai&2)  (ci2Cs)  -  (a^bs)  (a^c,) 

=  (ai^a)  (^2^3  —  cisGo)  —  (a^bs)  (ttjCa  — agCi) 
= ajj  [  (ai  62)  C3  +  (a2  ^3)  ^i]  -  Cg  [  (ai  62)  ^3  +  («2  ^3)  «i]  • 
The  last  binomial, 

(aj  62)  «3  +  (^2  ^3)  «1  =   («1  ^2  —  «2  ^1)  %  +  («2  &3  —  «3  ^2)  «!       - 
=  ^2  («1^3  —  «3^)  =  «2  («1^3)  • 

.-.  C  =  a2 [(ai62)  C3  —  (aibs)  Gi  +  («2&3)  cj 

=  ttg  [«!  &2  C3  —  ttg  ^1 C3  —  «i  &3  C2  +  ^3  61  Co  +  0^2  ?>3  Cj  —  Cl^  h^  cj  . 

Here  the  quantit}'  within  brackets  consists  of  2-3  =  G  terms, 
^.e.,  of  as  many  terms  as  there  are  permutations  of  the  sub- 
scripts 1/2, 3-  Three  of  the  terms  are  positive  and  as  many 
are  negative.     The  quantity  involves  the  3^  =  9  quantities  a^, 

^29    ^35   "H    b^-,    O3,   Cj,   C2,   C3. 

No  term  involves  more  than  one  a,  or  &,  or  c,  but  does  con- 
tain all  of  the  subscripts  1, 2, 3?  each  term  containing  a  different 
permutation  of  these  numbers.  Finally,  as  before,  we  notice 
that  those  terms  in  which  the  subscripts  occur  in  their  natural 
order,  or  in  which  there  is  an  even  number  of  inversions  *  of 

*  In  a  series  of  integers  which  are  all  different  there  is  said  to  be  an 
inversion  of  order  when  a  greater  number  precedes  a  less.  Thus  in  13452 
there  are  three  inversions,  in  21354  there  are  two  inversions,  etc. 


PRELIMINARY  NOTIONS   AND  DEFINITION^.  5 

order,  are  positive,  while  those  terms  are  negative  in  which  the 
number  of  inversions  of  order  of  the  subscripts  is  odd.  Such 
a  function  is  a  determinant  of  the  third  order.  A  determinant 
in  which  the  quantities  are  those  of  C  is  denoted  by  (rti&aCa). 
We  therefore  have  C  =  a2  (%&2C3) .  It  must  be  carefully  noticed 
that  the  equation 

(tti^gCg)  =  (a^bo)  Cs  —  (chbs)  c^  +  {ci^h)  Cj 

=  tti^aCs  —  a^hiC^  —  aibsC2  +  (Xs^iCa  +  ci2^sCi  —  a^b2Ci 

gives  the  expansion  of  a  determinant  of  the  third  order. 

Employing  the  notation  just  explained,  the  coefficient  of 
t  in  (1)  is  evidently  a^^aib^d^),  and  the  absolute  term  is 
«2(ai&2W3)«  The  coefficients  and  the  absolute  term  of  (2)  will 
obviously  be  a^Xaob^c^),  a-^ia^b^d^^  a^ia^b^m^^  in  order. 

Introducing  this  notation  into  set  IV,  and  dividing  (1)  and 
(2)  by  tta  and  a^  respectively,  we  have 

( 1 )    (ai  b. C3)  ^  +  (ai  62 ^^3)  ^  =  («i h m^)  , 


;;i 


( 2 )    (ttg 63 C4)  2;  +  (as &3 <^4)  t=  (a2bs m 

6.*   If  we  now  eliminate  z  in  the  same  manner  as  heretofore, 
we  have 


=   («1&2C3)   («2&3W^4)   —   («2^3C4)    («1&2^%) 


VI. 


The  preceding  results  naturally  imply  a  simplification  and  law 
of  formation  to  be  discovered  in  the  coefficient  of  t  and  the 
absolute  term  of  VI. 

To  simplify  the  coefficient  of  f,  which  for  shortness  we  will 
call  (7,  as  before,  we  proceed  as  follows : 

C  =  (ai^gCs)   ((1263^4)   —  (<^2&3C4)   (0tl&2^3) 

=  (aihcs)  [(a2^3)  d*  —  («2M  <^3  +  («3&4)  (^2] 

-  ((hhCi)  [(Ctj  62)  ^3  -  (CLlps)  ^2  +  («2&3)  <^J 

=  {<hh)  [(ai&2C3)  <^4  -  ( o.ihc^)  d{\  -  Dd.,  +  D,d2', 
*  6  may  be  omitted  on  first  reading,  if  thought  best. 


6  THEOHY   OF   DETERM1NA^'TS. 

in  which 

D  =  {a^h^c^  (^2^4)  +  (a2^3C4)  (%&2)?  and 

Now,  by  5,  (aidgc's)  =  {aib^)  Cg  —  {aM  c^  +  {a^h)  Ci ; 
and  (ag  63 C4)  =  (ag  ^3)  ^4  —  («2  ^4)  C3  +  (t^s  ^4)  ^2- 

Substituting, 

D  =  (a^h)  [ia,\)  C3  -  (ai&g)  Cg  +  (agftg)  cj  +  (aiftg)  [(«2&3)  C4 

—  («2MC3+(«3^4)C2] 
=  (as 63)   [(^2^  Ci+  (Oti^s)  Cj  -  [(a2&4)   {(^ih) 

-{a^h)  (a3&4)]c2- 

The  second  binomial,  {a^h^)  {aih^)  —  ((1162)  (^3^4) 

=  {a^h^-a^\)  {aM  -  (^3^4 -«4 ^3)  ((^ih) 

=  &4  [(ai?>3)  0(2  -  (oti&2)  «3]  -  <^4  [(tti^s)  ^2  -  (^1^2)  hi 

=  ^4  [(«1^3  —  «3&l)  «2  —  («1^2  — «2^)  ^3] 

—  ^4  [(tti 63  —  ttg 61)  62  —  (<*! h  —  a^bi)  63] 
=  <Xi  64  (as 63  —  cca  bo)  —  O4 61  (tta ^s  —  %  ^2) 

=  (0164)  (ag&g) (K) 

.*.  D  =  {a^b^)  [(ai^a)  C4  —  (ai&i)  ^2  +  (a2&4)  cj. 

Substituting  the  expansions  of  {a^b^c^)  and  of  {a2biC^  in  Di 
we  have 

A  =   («'3M   [(«1^2)  C3  -  (ai?>3)  ^2  +  («2W  Cj  +  («l&3)   [(a2^3)  C4 

—  («2WC3+  («3MC2] 

=  {a.p.,)  [(a3&4)ci+  (0163)^4]  -  [(ai&3)  («2?>4)  -  («A)  (ot3^4)]e3. 

Here  we  notice  that  the  binomial  factor  of  the  second  term  is 
the  same  as  the  binomial  factor  in  the  last  term  of  D :  hence 
equation  (K)  above,  is  {a^^b^  (<^i&4)« 

.-.  A=   {(^ih)  [(«1^3)C4-  («1&4)C3+  (a3^4)^i]. 

Substituting  the  values  of  D  and  Di  just  obtained,   in  C, 
we  have 


PRELIMINARY  NOTIONS    AND   DEFINITIONS.  7 

O  =  {a^h)  [_{a^h.2C^)d^—  {a^h^c,)^.^—  \{(hh.?)c^~  («! 64)03 

=  (^^s)  l{.(^ihG^)d^—  (a^hc^)d.i-h  {chb^Ci)(l2—  (aobsC,)d{]. 

From  this  value  of  C  the  absolute  term  of  VI  is  obviously 

((^kh)  l(ctib2C^)mi—  (ai 62^)^3+  (^i 63 04)^13—  (a2&3C4)mJ. 

Now  the  quantit}^  within  brackets  in  C  (and  in  the  absolute 
term)  of  YI  is  here  seen  to  be  composed  of  four  terms,  each 
of  which  contains  a  factor  which  is  a  determinant  of  the  third 
order.  We  shall  presently  show  that  this  quantity  is  a  deter- 
minant of  the  fourth  order ^  and  will  therefore  write,  in  accord- 
ance with  the  notation  already  exemplified,  for  determinants  of 
lower  orders  : 

{a^h2C^)d^  —  {a^b.2C^)ds  +  («iV4)<^2  —  (ct2hcd^i  =  (<^hhc3d4)  •  •  •  (R). 

Now,  5,        («!  62  Cs)  =  («!  h)  C3  -  («!  63)  6-2  +  (a2  63)  Ci ; 

(aib2C^)  =  (ai  62)04  —  (%  64)02  +  (^2  64)^1 ; 

(ciibsc^)  =  (ai 63)04  -  (0^^64)03+  ((1364)01 ; 

(a2  6304)  =  («2  63)04—  (0^264)03+  (a3  64)02. 

Expanding  the  determinants  of  the  second  order  in  the 
second  members  of  these  equations  according  to  4,  and  sub- 
stituting in  equation  (R) ,  there  results  ; 

(ai6203cZ4)  =ai6203d4— a26i03cZ4— a.i6302f^4H-  a36i02d4+  a2b^Cid^— a^^^^id^ 
— aib^c^d^ + a^biCids + a^^c^d^  —  a^-^c^d^  —  a'^^c-^d^ + ap^^^d^ 
+(116304(^2— «36i04C^2—«i64C3^2+  a46i03d2H-  a^b^pid^—  a^b^c^d^ 
—  a2bzC4il^-\-a^h^^di-\-a2b4p^di—  a^2f-'.^di—  a^b^c^di-^  ajft-^c^di. 

This  expansion  contains  4 -S- 2  =  24  terms,  involving  4^=16 
quantities.  Each  term  contains  only  one  a  (coefficient  of  x) , 
or  6  (coefficient  of  ?/) ,  or  0  (coefficient  of  z)\,  or  d  (coefficient 
of  t) ,  and  contains  all  the  subscripts ;  a  different  permutation 
of  the  subscripts  belonging  to  each  term.  As  before,  we  find 
that  the  number  of  inversions  of  order  of  the  subscripts  is  an 
even  number  in  the  posi^ve  terms,  and  is  an  odd  number  in 


8  THEORY  OF  DETERMINANTS. 

the  negative  terms.  Moreover,  the  number  of  terms  is  exactly 
the  number  of  permutations  of  the  first  four  natural  numbers. 
Such  a  function  is  a  determinant  of  the  fourth  order,  and  is 
accordingly  designated  by  {aih^c^d^.  Introducing  this  nota- 
tion, and  dividing  by  (aa^s)?  equation  VI  becomes 

{aih2Czd^t=^  (aibzCsm^),     VII.  ' 

It  is  to  be  noticed  that  equation  (R)  of  the  present  article 
gives  the  expansion  of  a  determinant  of  the  fourth  order. 

7.  We  have  now  shown  how  determinants  of  the  second, 
third,  and  fourth  orders  arise  in  the  solution  of  simple  simul- 
taneous equations.  From  the  reductions  of  6,  it  is  obvious 
that  to  continue  the  present  method  would  very  soon  imply 
difficulties  in  the  simplifications  practically  insurmountable  when 
we  attempt  to  produce  determinants  of  the  higher  orders.  For 
determinants  of  the  fifth  order,  the  process  of  reduction  would 
be  found  very  tedious.  Hence,  to  investigate  the  properties  of 
determinants  of  the  nth  order,  we  are  forced  to  take  a  new 
starting-point ;  and  in  Chapter  II.  we  proceed  upon  a  plan 
somewhat  different  from  that  hitherto  adopted. 

Values  of  the  Unknown  Quantities. 

8.  From  equation  VII,  6,  ^  =  T-^-v-^rv-    Had  the  equations 

of  set  I  been  so  arranged  that  z  should  be  the  last  unknown 

m  each  equation,  we  would  evidentlv  have  z  =  -y — 7—5 — r- 
^    ,,  (a,d,c,m,)     ^     ^(d,b,c,m,)    (^^^^ds^^) 

In  the  same  way,  y  =  ^^-^^-j ;    x  =  "p^^^^j- 

9.  Among  the  many  properties  of  determinants  to  be  estab- 
lished, we  may  here  produce  the  following  theorem,  which  is 
among  the  most  important  of  the  elementary  theorems  in  the 
subject : 

The  interchange  of  two  letters,  or  of  two  subscripts,  the  others 
remaining  undisturbed,  changes  the  sign  but  not  the  magnitude 
of  a  determinant.  ^ 


VALUES    or    THE   UNKNOWN    QUANTITIES.  9 

1st.    For  determinants  of  the  second  order. 

(a)  The  interchange  of  two  letters. 

{aih.2)  =ai62  — ^2^1-  In  this,  if  we  interchange  a  and  5,  the 
second  member  becomes 

hia^  —  b2ai=  —  (ai&2  —  «2^i)  •*•  (^i<^'2)  =  —  (f^i^a)  • 

(b)  The  interchange  of  two  subscripts. 

{a^h^  =ai62  — ^^2^1'  If  the  subscripts  are  interchanged,  the 
second  member  becomes 

a^hx  —  a-Jy^  =  —  («i?>2  —  «2^i)  •*•  (^2^1)  =  —  (^^2)  • 

2d.    For  determinants  of  the  third  order. 

(a)  The  interchange  of  two  letters. 

(tti^aCs)  =  («i^2)c3—  {(^\bz)<^2+  («2^3)<^i-  In  this,  if  we  inter- 
change a  and  &,  the  proposition  is  obvious  from  the  first  part 
of  the  demonstration,  (a). 

We  have  therefore  to  show  that  the  proposition  holds  for  b 
and  c.     We  have,  5, 

{aib.^  {a^c^)  —  {a^b^)  {axG,^)^  a.{o-ih<^8) - 

In  this  expression,  interchanging  b  and  c,  the  first  member 
becomes  {a-^c^)  (^^s)  —  {(^'2<^z)  (<^hh)'  Since  a^  remains  un- 
changed, (aiCg&s)  =  —  (r^i^oCs). 

(b)  The  interchange  of  two  subscripts. 

(oi 60 C3)  =  (tti 62)^3— (^1^3) <^2+ («2 ^3)^1-  (I')-  If  the  sub- 
scripts 2  and  3  are  interchanged,  the  second  member  becomes 
{aibs)c2—  (ai&2)c3+  («3&2)ci.  Since  {cisbo)  =  —  («2^3)?  1st, 
(6) ,  the  second  number  of  (L)  becomes 

—  (ai&2)c3+  {aibs)Co—  (a.2bs)ci 

In  the  same  manner  it  may  be  shown  that  the  interchange  of 
any  other  two  subscripts  in  (L)  changes  the  sign  of  the  second 
\      member,  .*.  the  proposition. 


10  THEORY   OF   DETERMINANTS. 

3d.    For  determinants  of  the  fourth  order. 

(a)  The  interchange  of  two  letters. 

(aiftgCa^O  =  {a^hcz)d^—  {a^hc^d^-^  {aih^c^d^—  (a2&3<J4)^i    (I*)- 
From  2d,  (a) ,  the  proposition  is  obvious  for  an  interchange 
of  the  first  three  letters.     To  show  that  the  proposition  holds 
for  c  and  d,  we  have,  6, 

The  interchange  of  c  and  d  transforms  the  minuend  into  sub- 
trahend, and  the  subtrahend  into  minuend,  in  the  first  member. 
Hence,  as  (ag^s)  remains  unchanged,  {aib2d3C4)  =  —  {aiboC^di) . 

(b)  The  interchange  of  two  subscripts. 

{a^b^c^d^)  =  {aAcs)  d^  —  (aib2C^)  d^  +  (a^hc^) d^  —  {aihc^ d^.    (M) . 

In  this,  if  we  interchange  the  subscripts   2  and  3?  the  second 
member  of  (M)  becomes 

(ai  63  C2)  (^4  —  («!  63  C4)  (^2  +  (oti  62  C4)  do,  +  (%  &2  C4)  di. 

Now,  by  2d,  (6),  {aib^c^)  =  —  (aib^Cs) ;  and  (ag&aQ)  =  —  (ot2&3C4). 
Hence  the  second  member  of  (M)  may  be  written 

—  (tti  &2  C3)  d^  +  (ai  62  C4)  ds  —  («!  &3C4)  ^2  +  (02  &3  C4)  di, 
and  therefore  (ai 6302^4)  =  —  (ai b2C3d^) . 

In  a  similar  manner  the  proposition  may  be  established  for 
the  interchange  of  any  other  two  subscripts. 

It  is  obvious  that  two  consecutive  interchanges  will  leave  the 
determinant  unaltered  either  in  sign  or  magnitude.  Notice 
that  an  interchange  of  two  letters  corresponds  to  a  uniform 
change  in  the  order  of  succession  of  the  unknown  quantities  in 
the  original  set  of  equations.  Also,  that  an  interchange  of  two 
subscripts  corresponds  to  changing  the  order  of  the  equations. 

10.  Applying  the  proposition  of  the  preceding  article  to  the 
values  of  cc,  y^  z,  and  t,  obtained  in  8,  we  have 

__  (mi 62^3 0^4)  ,     _{0im2C^di)  ^     _(ai?)2^3<^4).  .  _  ((^ib^c^m.i) 
"■  (% 62^3^4)  '  {dibiCsd^)  '     ~  (ai&2C3C?4)'     ~  {ctibzC^d^)' 


NOTATION.  11 

Notice  that  the  common  denominator  in  these"  values  is  the 
determinant  of  the  fourth  order,  formed  from  the  coefficients  of 
the  unknown  quantities.  Also,  that  the  numerator  of  the  value 
of  X  is  obtained  by  changing  the  a  of  the  denominator  into  m. 
The  numerator  of  the  value  of  y  is  likewise  obtained  by  chang- 
ing the  b  of  the  denominator  into  m,  and  that  the  numerators 
of  the  values  of  z  and  t  are  similarly  obtained  b}^  changing  the 
c  and  d  into  m  respectively. 

Notation. 

11.  We  have  seen  that  a  determinant  of  the  second  order 
contains  2^=4  quantities,  a  determinant  of  the  third  order 
3^  =  9  quantities,  and  a  determinant  of  the  fourth  order  4^  =  16 
quantities.  It  is  customary  to  employ  the  notation  introduced 
b}-  Cayley,  and  write  these  determinants  so  that  the  quantities 
(called  elements)  entering  into  the  determinant  appear  arranged 
in  the  form  of  a  square,  with  a  vertical  line  on  each  side. 


Thus  (aib2)=  cti^i  ;  (aibzCs)^  aybyCi  and  (aj 62^3(^4)=  aibiCidi 

02^2  ^<2^2C2  a2b2C2d2 

a^b^Cid^ 

Other  forms  of  notation  are  also  |  cti  62 1  for  (aj  62) ;  |  ai  62  Cg  | 
for  (0162^3);   |ai&2C3<^4l  ^ov  {aib2Csdi). 

There  are  still  others  to  be  described  later.  In  Cayley's 
notation  the  elements  are  so  arranged  that,  regarded  as  coeffi- 
cients of  the  unknowns  in  the  original  set  of  equations,  they 
occur  in  rows  and  columns  in  the  regular  order  in  which  they 
are  found  in  these  original  equations.  Further,  comparing  the 
expansions  with  the  square  arrangement,  we  notice  that  each 
term  contains  one,  and  only  one^  element  from  each  row  and 
cohimn,  and  that  there  is  no  other  element  from  the  same  row 
and  column  in  the  same  term.  Hence,  as  already  exemplified, 
there  can  be  only  2,  3,  or  4  elements  in  each  term,  according 
as  the  determinant  is  of  the  second,  third,  or  fourth  order. 
It  will  be  noticed  that  the  quantities  occurring  in  the  abbreviated 


12 


THEony  or  determinants. 


forms  (oihoCo),  (ciiho^),  \a1b.2 0^(1^1,,  etc.,  are  those  found  in  one 
of  the  diagonals  in  the  square  arrangement,  viz.,  the  diagonal 
extending  from  the  upper  left-hand  corner  to  the  lower  right- 
hand  corner.  This  diagonal  is  called  the  principal  diagonal. 
Similarly,  that  diagonal  extending  from  the  lower  left-hand 
corner  to  the  upper  right-hand  corner  is  the  secondary  diagonal. 
Any  line  parallel  to  these  (principal  or  secondary)  is  a  minor 
diagonal.  Any  of  the  expansions  heretofore  given  show  that 
the  product  of  the  elements  of  the  principal  diagonal  is  a  posi- 
tive term  of  the  determinant.  This  term  being  composed  of 
the  elements  of  the  principal  diagonal,  is  called  the  principal 
term.  The  other  terms  can  be  formed  from  the  principal  term 
by  making  all  the  possible  permutations  of  the  subscripts  and 
prefixing  the  proper  sign  to  each  permutation  (5  and  footnote  ; 
also  6). 

Observe  that  the  order  of  the  letters  in  the  abbreviated  forms 
of  notation  is  the  order  of  the  columns  in  the  square  arrange- 
ment, and  that  the  order  of  the  subscripts  gives  the  order  of 
the  rows.  Thus,  [ai^gCsl  means  the  determinant  whose  first 
column  consists  of  a's,  second  column  of  &'s,  and  third  column 
of  c's,  and  that  the  subscript  of  each  letter  in  the  first  row  is  1, 
and  in  the  second  each  letter  has  the  subscript  2?  and  in  the 
third  each  letter  has  the  subscript  3. 


Illustrations  are : 

|a3&2C4|   = 

«3  h  C3 
a.2  b.,  Co 
04  64  C4 

' 

1036405(^1!  = 

V 

«3  63  C3  d^ 
a^h^c^d^ 

«'6  h  C5  d^ 
Oi  hi  Ci  di 

;  («i 

C2bs)  = 

Oi  Ci  61 
a2C2&2 

030363 

\a,h,c^d,\  = 

Oy  b^  c^ 
a.  b,  c, 

a,b,c. 

dr 

d. 
d. 

• 

' 

♦• 

^*^iVE:^s\\ 


i£lLIFORN\^ 
NOTATION. 


13 


The  expansion  of  determinants  of  the  second  and  third  orders. 

12.  Though  we  have  alread}'  given  the  expansion  of  deter- 
minants of  the  second  and  third  order  several  times,  it  will  be 
useful  here  to  compare  these  expansions  with  the  square 
arrangement  once  more.  Also,  we  are  now  prepared  for  a 
convenient  mnemonic  rule  for  the  expansion  of  a  determinant 
of  the  third  order,  to  be  given  in  15. 

13.  Since  I  «i&i    =ai&2  — ^2^15  it  is  obvious  that  the  expan- 

I  Ct2  &2 

sion  of  a  determinant  of  the  second  order  is  obtained  by  taking 
the  product  of  the  elements  of  the  principal  diagonal  and  the 
product  of  the  elements  in  the  secondary  diagonal,  and  sub- 
tracting the  second  product  from  the  first. 

14.  We  have  repeatedly  shown  that 

Ci. 


ttl  61  tl 

= 

ai6i 

C3- 

a,b. 

C2  + 

aa^a 

Osf&gCa 

0^2  62 

a^h 

«3&3 

agftgCg 

From  this  it  appears  that  a  determinant  of  the  third  order  can 
be  decomposed  into  determinants  of  the  second  order,  each 
multiplied  by  the  elements  in  order  of  the  last  column,  begin- 
ning with  the  last  element.  Since  any  column  may  be  made 
the  last,  9,  the  assertion  just  made  amounts  to  saying  that 
a  determinant  of  the  third  order  may  be  expressed  in  terms 
of  determinants  of  the  second  order  and  the  elements  of  any 
column. 

The  reader  will  readily  see  how  the  determinant  factors  of 
the  expansion  in  the  present  article  are  obtained  from  the 
original  determinant.  For  example,  the  ciifactor  of  Cg  is  ob- 
tained by  striking  out  the  row  and  column  in  which  c^  is  found, 
and  regarding  what  is  left  as  a  determinant  of  the  second  order. 
Thus,  I 

(h  ^3  ?3 


14 


THEORY  OF   DETERMINANTS. 


15.  The  following  convenient  rule  for  the  complete  expan- 
sion of  a  determinant  of  the  third  order  is  indicated  in  the 
accompanying  diagram,  and  is  described  as  follows :  — 

The  terms  composed  of  ele- 
ments of  the  principal  diagonal 
and  of  the  minor  diagonals 
parallel  to  it  are  positive,  while 
those  formed  of  elements  in 
the  secondary- diagonal  and  the 
minor  diagonals  parallel  to  it 
are  negative.  The  elements 
pierced  by  the  double  lines 
compose  the  positive  terms. 
The  elements  pierced  by  tlie 
single  lines  similarly  consti- 
tute the  negative  terms.  In  accordance  with  these  directions, 
the  expansion  of 


6^2  0.2  C^ 
«3  h  ^2, 


is  ai^a^'sH-  «2^3Ci  +  da^iCg  —  ag^gCj  —  aa&iCg  —  aih^C2. 


This  is  identical  with  the  expansion  already  obtained  in  5, 
it  should  be. 


as 


EXAMPLES. 


1 .    Find  the  values  of  : 


4  -G 

5  3 


25   18 
49   75 


10  -6 
8   -3 


a  b 
b  a 


7  1 
5  0 


2.    Write  in  determinant  form  : 


a-\-  b  b 
a  -\-b  a 


0  3 
0  4 


0  1 


a  -b  b\; 
b  —c  c  \ 
1 
a 


7  ;  5  :  16  ;  —13;Xiy  —  xy',  3a  — 7b  ;  (^  —  bd; ;  Sgh  —  xy. 

b      a 

3  -1 

1      2 


(Suggestion  :— 7  =  3x2  —  (Ix— 1) 


Numberless 


other  forms  could,  of  course,  be  given  for  the  same  quantity.) 


EXAMPLES. 


15 


3.    Without   passing    from  the  determinant   notation,   show 
what  relation  exists  between 


(9.) 


a,  b. 

and 

b^ai 
62  ^2 

.     Also 

X  y 
m  n 

and 

m  n 
X  y 

• 

Compare 

a  b 
c  d 

and 

a  c 
b  d 

Also  < 

3ompa 

re 

a  b 
G  d 

5     ' 

3a 

c 

36 
d 

,  and 

3a 
3c 

b 
d 

5.  Write  the  expansion  of  the  following  determinants  : 
{a^h);   (a^bj,);  \a^biG„,\;  {a^b^CQl;   (ag  65  Cj) ;    162^3^1 

6.  Find  the  values  of  : 


12  3 

5 

4  7  8 

5 

a  0  0 

? 

0  0a 

1 

a  0  c 

5 

a  6  c 

4  5  6 

0  3  6 

0  6  0 

b  c  0 

6  0  6 

b  c  a 

7  8  9 

0  5  9 

0  0  c 

0  0  6 

c  0  a 

cab 

7.    Compare 


Also  compare 


Also  compare 


8.    State  the  probable  theorems  exemplified  by  the  results 
Ex.  7. 


a  0  c 
d  0  f 
g  Ok 

and 

a 
d 
9 

0  0  . 
h  k 

a  mb  c 
d  me  f 
g  mh  k 

and 

a  b  c 
d  e  f 
g  h  k 

tti  61  Ci 

a.2  62  C2 

ttg  63  Cg 

and 

a 
6 
c 

1^2  «3 

1  ^2  h 

1    ^2    C3 

9.    Find  the  value  of  x  in  the  equations : 


(1) 


(3) 


X      2 

= 

3  4 

;    (2) 

1  -1 

5  6 

4  1 
3  -2 
2       1 


=  0 


1 

1 

1 

a 

X 

c 

6 

6 

X 

=  0;   (4) 


X  a  a 

=  — 

6  6  a; 

a  X  a 

6  a;  6 

a  a  X 

a;  6  6 

16 


THEORY   OF  DETERMINANTS. 


10.    Find  the  complete  expansion  of 

=  I  a^  6<  c„  cZ^I .     (6,  equation  (R) ,  et  seq.) 


a. 

K 

Crd^ 

a, 

b, 

Ct  d, 

«» 

K 

dn^n 

a. 

\ 

Cod, 

11.  Wiite  in  determinant  form,  square  notation: 

(1)   bfg-\-  eid  -\-  lick  —  hfd  —  ecg  —  bik. 

(2  )  Till  Tig  rg  —  my  %  rg  +  mg  n^  r^  —  m^  Ui  r^  +  mg  Wj  ?'2  —  WgTigri. 

(3)   Sxyz-a^-f  —  ^. 

12.  Employ  9  to  compare  the  following : 


a  b  c 
d  e  f 
g  h'k 

and 

d  e  f 
g  h  ^k 
a  b  c 

'i 

m  n  0 
p  q   r 

s    t   u 

and 

0  n  m 
r  q  p 
u  t   s 

m  n  0 

and 

0  m  n 

, 

p  q  r 
s    t   u 

r  I 

U    i 

0  q 
f   t 

13.  Expand  the  following  in  terms  of  determinants  of  the 
second  order  and  the  elements  of  any  column  (14).  Verify 
the  results  by  making  use  of  the  rule  in  15 : 


«!  6i  Ci 

1 

002  2/2  ^2 

5 

a^  64  C4 

CL2  bo  C2 

^*3  2/3  '^h 

as  b.  C3 

«3  h  Cs 

Xi  y^  m^ 

C/6  ^6  Cs 

14.    Count  the  inversions  of  order  in 

(a)  13  5426  7 

(6)  2  3  6  1457 

(c)  6  3  5  4  17  2 

(d)  789653421 

(e)  987654321 


CHAPTER  II. 

GENERAL  PROPERTIES  OF  DETERMINANTS. 
"^  Notation  and  Definition. 

16.  The  investigations  of  the  preceding  chapter  have  revealed 
the  fact  that  a  determinant  of  the  second,  third,  or  fourth  order 
is  a  function  of  2^,  3^,  or  4^  quantities  respectively,  and  have  also 
established  a  uniform  law  of  formation  for  these  functions.  In 
order  therefore  to  investigate  the  properties  of  Determinants  in 
general,  we  have  but  to  consider  a  function  of  ?i^  quantities 
whose  law  of  formation  is  given  in  the  following  definition. 

17.  Definition.  —  A  Determinant  is  always  a  function  of  n^ 
quantities.  These  quantities,  called  elements,  being  arranged 
in  the  fonn  of  a  square  consisting  of  n  rows,  and  thus  also  of 
n  columns,  n  quantities  in  each  row  and  in  each  column,  the 
determinant  of  these  n^  quantities  is  the  sum  of  the  terms 
formed  as  follows :  *  Each  term  is  the  product  of  n  elements, 
so  chosen  that  there  is  one  element  from  each  row  and  one 
from  each  column,  —  but  two  elements  from  the  same  row  or 
column  must  never  occur  in  any  one  term.  The  sign-factor  of 
each  term  is  (— 1)^+',  in  which  p  is  the  number  of  inver-  / 
sions  of  order t  of  the  rows,  and  i  is  the  number  of  inversions  > 
of  order  of  the  columns,  from  which  the  elements  composing 
the  term  have  been  chosen. 

Note.  —  Each  term  being  composed  of  n  factors,  the  deter- 
minant is  said  to  be  of  the  nth  order  or  degree. 

*  22  et  seq.  will  show  that  the  law  of  formation  given  in  this  definition 
is  the  same  as  that  already  observed  in  determinants  of  the  2d,  3d,  and 
4th  orders  (3  to  6  inclusive). 

t  5,  footnote  on  inversions  of  order. 


18 


THEORY   OF   DETERMINANTS; 


18.    To  expand 


a 

b  c 

d 

ef 

mn  o\ 

by  the  definition,  we  may  select  any 


row,  as,  for  instance,  the  second  row,  and  using  each  element* 
of  that  row  in  turn,  according  to  the  directions  given,  we  shall 
form  all  the  terms  of  the  determinant.  For  the  first  term,  then, 
taking  d  as  the  first  element,  we  see  that  we  can  take  b  and  o 

iy,  h    c 

-M  n  0 

for  the  other  factors  of  a  term,  and  no  more,  since  we  have 
then  chosen  one  element  from  each  row  and  one  from  each 
column,  and  no  two  elements  are  from  the  same  row  or  column. 
We  now  have  the  term  dbo.  To  form  another  term  containing 
d,  we  can  evidentlj'  take  n  and  c,  giving  the  term  dnc^  which  as 
before  contains  an  element  from  each  row  and  column,  and  no 
two  elements  are  from  the  same  row  or  column.  No  other 
terms  containing  d  can  be  formed.  The  terms  containing  e  are 
in  the  same  way  eao  and  mec ;  the  diagram  will  sufficiently 
explain  the  manner  of  obtaining  these  terms. 

a  '1}^  c 

m  i^  0 
The  terms  containing  /  are  likewise  naf  and  fbm. 


a  1)  ^ 
m  n  6 


To  fix  the  signs  of  these  terms,  we  will  write  under  each 
term  the  numbers  giving  the  rows  and  the  numbers  giving  the 

*  There  is  a  difference  in  the  nomenclature.     What  we  have  called 
elements  some  authors  call  constituents,  and  an  element  is  a  term. 


GENEKAL   PllOPERTIES    OF    DETERMINANTS. 


19 


columns  from  which  the  elements  have  been  taken,  and  opposite 
eacli  series  the  number  of  inversions.     Thus  : 

dho  dnc  eao  mec  naf  fhm 

Rows  213-1  231-2  213-1  321-3  312-2  213-1 
Columns  123-0  123-0  213-1  123-0  213-1  321-3 
The  sum  of  the  inversions  of  order  in  rows  and  columns  of  the 
first  term  is  unity;  .-.  (— 1)^=  — 1,  and  dho  is  negative.  In 
dnc  the  sum  of  inversions  of  order  in  rows  and  columns  is  2  ; 
.-.(—1)^=1,  and  dnc  is  positive.  Similarly  for  the  other 
terms.     Affecting  the  terms  with  their  proper  signs, 


a  h  c 
d  e  f 
m  n  o 
Scholium.  - 


dho  +  dnc  -f  eao  —  mec  —  naf  -\-  fhm. 


This  illustration  is  inserted  only  to  give  the 
reader  a  clear  idea  of  the  meaning  of  the  definition,  and  not 
'|*>  because  we  really  employ  the  definition  in  the  practical  expan- 
sion of  determinants.  In  fact,  the  great  beauty  of  the  deter- 
minant notation  is  that  we  are  able  to  conduct  most  of  our 
investigations  with  the  help  of  determinants  without  requiring 
the  expansions  at  all.  In  case  it  becomes  necessary  to  expand 
a  determinant,  we  have  several  excellent  methods  to  be  given 
later.  One  method  for  the  expansion  of  a  determinant  of  the 
third  order  has  been  given  already  (15) . 

19.    In  accordance  with  the  notation  already  exemplified  in 
Chapter  I.,  a  determinant  of  the  nth  order  is  written 
tti  hi  Ci  ...  li 

O2  0-2  C2   ...    I2 

a^  hs  C3  ...  Is 


<.ln\,  or  to 


a„  6«  c„  ...  l„ 
This  form  is  shortened  to  (ajdaCa ... /„)  or  \a1h2C3 
2  ±  cti&aCg  ..*.  l^.  In  each  of  these  shortened  forms  those  ele- 
ments occur  which  occupy  the  principal  diagonal*  in  the  square 
arrangement.  The  form  S  ±  ai^a^s ...  ?„  is  suggestive  of  the 
manner  in  which  the  function  is  formed.     The  5  ±   stands  for 


*  11. 


20 


THEORY   OF    DETERMINANTS. 


the  sum  of  all  the  terms  that  can  be  formed  from  the  pnncipal 
term  b}'  permuting  the  subscripts  and  prefixing  the  proper  sign 
to  each.     (23.) 

Another  and  very  convenient  notation  is  obtained  by  employ- 
ing a  single  letter  affected  with  two  subscripts ;  the  first  sub- 
script giving  the  row,  and  the  second  subscript  the  column,  in 
which  the  element  occurs.     Thus  : 

ail  ^12  ^%  •••  %» 

OE'21  tl22  <^23  •••   ^2w 
a^i  a^2  <^33  •••   <^3» 


This  form  may,  like  the  first,  be  shortened  to  |  «ii  «22  •••  ct„„|, 
(ciii  a22  ^33  •  •  •  ^nn) ,  or  2  ±  «!!  «22  «33  •  •  •  «„»!•  It  may  also  be  still 
further  abbreviated  to  |  ai^  | .  A  modification  of  this  notation, 
with  the  two  subscripts,  consists  in  omitting  the  letter  alt^j 
gether,  and  writing  the  determinant  thus  : 


(1,1)  (1,2)  (1,3)  ...  (l,n) 
(^2,1)  (2,2)  (2,3)  ...  (2,n) 
(3,1)  (3,2)  (3,3)  ...  (3,71) 


(n,l)  (n,2)  (n,3) 
or,  finally,  /I  2  3 


,.  (n,n) 


or 


11 

12  13  . 

.   In 

21 

22  23  . 

.  2n 

31 

32  33  . 

.     3  71 

nl 

n2  7i3  . 

..    7171 

',  /I  2  3  ...  7iY 
1,1  2  3  ...  n) 


These  last  three  forms  are  called  the  iimbral  notation. 

20.  The  following  corollaries  flow  from  the  definition  in  17. 
They  are  obvious  upon  a  moment's  reflection. 

Cor.    I.  — The  principal  term  is  always  positive. 

Cor.  II.  — If  each  element  of  a  row  or  of  a  column  is  zero, 
the  determinant  vanishes. 


^^  General  Properties. 

21.    Theorem.  —  If  in   a   series   of  integers  tvhicJi   are 


aJI 


different^  any  two  are  interchanged^  the  others  remaining  undis- 
turbed, the  number  of  inversions  of  order  is  thereby  increased  or 
diminished  by  an  odd  number* 


GENERAL  PROPERTIES  OF  DETERMINANTS.     21 

Let  the  series  of  integers  be  Ae  Bf  (7,  in  wliich  A  is  used  to 
denote  the  series  ay/c ...  preceding  e,  B  denotes  the  series  hgl ... 
between  e  and  /,  and  G  the  series  following  f. 

In  the  first  place,  it  is  evident  that  if  any  two  adjacent 
integers  are  interchanged,  the  number  of  inversions  of  order 
is  thereby  increased  or  diminished  by  unity.  For  let  vm  be 
any  two  adjacent  integers  in  a  series.  ^If  we  write  mv,  we 
introduce  one  inversion  of  order  if  m  >?;.  Or,  if  m<v,  we 
have  lost  an  inversion.  Now,  since  this  change  cannot  affect 
the  rest  of  the  series,  we  have  increased  or  diminished  the  total 
number  of  inversions  in  the  series  by  unity. 

Again,  in  order  to  interchange  e  in  Ae  Bf  (7,  with  /  separated 
from  e  by  7t,  intervening  elements,  we  may  first  interchange  e 
with  the  elements  to  the  right  in  regular  succession  k-\-l  times  ; 
I  this  brings  e  into  the  place  at  first  occupied  by  /.  Then,  in 
order  to  transfer  /  to  the  place  formerly  occupied  by  e,  we  have 
to  pass  /  over  k  elements  to  the  left.  Altogether,  we  have 
changed  the  *number  of  inversions  of  order  from  odd  to  even, 
or  from  even  to  odd,  2k-\-l  (an  odd  number)  of  times.  Hence 
the  proposition. 

22.  Theorem.  —  The  number  of  terms  in  a  determinant  of 
the  nth  order  is  1  -2  '2>  •  ...  n  =  n\ 

The  simplest  wa}'  to  form  the  terms  of  a  determinant  accord- 
ing to  the  definition,  is  to  choose  the  elements  from  the  columns 
in  order ;  that  is,  the  first  element  of  a  term  from  the  first 
column,  the  second  element  from  the  second  column,  etc. 
Choosing  the  elements  in  this  wa}',  we  may  take  the  first  ele- 
ment of  a  term  from  the  first  column  and  third  row,  say,  the 
next  element  from  the  second  column  and  any  row  except  the 
thirds  the  next  element  from  the  third  column  and  any  row 
except  those  already  selected,  and  so  on,  until  all  the  columns 
and  rows  have  been  drawn  upon.  The  numbers  of  the  rows 
from  which  the  elements  are  chosen  will  constitute  a  permu- 
tation of  the  numbers  1,  2,  3,  ...  n,  and  at  is  obvious  that 
we   can  therefore  select   the  elements  to  form  a  term  in  as" 


22  THEORY   OF  DETERMINANTS. 

many  different  ways  as  there  are  permutations  of  the  first  n 
numbers,  that  is  n  !     There  are  accordingly  n  !  different  terms. 

23.  Cor.  I. —  The  terms  of  a  detenninant  |  aidgCg  ...  ?„  I 
may  all  be  obtained  by  keeping  the  letters  in  alphabetical  order 
(^.e.,  choosing  the  elements  for  each  term  from  the  columns  in 
order),  making  all  the  possible  permutations  of  the  subscripts, 
and  prefixing  the  sign  +  or  —  to  each  permutation,  according 
as  the  number  of  inversions  of  order  is  even  or  odd.  Since 
the  expansion  of  a  determinant  in  accordance  with  the  definition 
would  also  be  obtained  by  keeping  the  rows  in  order,  and 
choosing  the  elements  from  the  columns  in  all  possible  ways. 
all  the  terms  of  |  ai  62C3  ...  Z„  |  can  be  formed  by  permuting  the 
letters,  keeping  the  subscripts  in  order,  and  prefixing  the  sign 
+  or  —  to  each  permutation,  according  as  the  number  of  in- 
versions of  the  letters  is  even  or  odd. 

24.  Cor.  II.  —  Similarly,  the  terms  of  |  a-^^  \  can  be  formed 
by  making  all  the  jjossible  permutations  of  the  first  set  of  sub- 
scripts and  keeping  the  second  set  in  order ;  or  the  terms  may 
be  obtained  by  making  all  the  possible  permutations  of  the 
second  set  and  leavinsr  the  first  set  in  order. 


Illustrations :  To  expand 


,  we  may  write  the  permu- 


ctj  hi  Ci 

(^2  ^2  ^2 
%  O3  C3 

tations  of  the  subscripts  in  a  column,  and  indicate  the  number 

of  inversions  of  order  in  each  by  a  figure  placed  at  the  right ; 

or  we  may  write  the  permutations  of  the  letters  in  the  same 

way.     Thus : 

1  2  3  ...  0  ah  c  ...  0 

1  3  2  ...  1  ach  ...I 
3  1  2  ...  2  6  a  c  ...  1 
3  2  1  ...  3  5  ca  ...  2 

2  3  1  ...  2  ca  6  ...  2 
2  1  3  ...  1  c  &  a  ...  3 

The  two  expansions  are  accordingly 

a^hoC^s  —  ai&gCo  +  ciz^iCi  —  ct^hoCi  +  aa^^aCi  —  cTo^iPsj 


GENERAL  PROPERTIES  OF  DETERMINANTS. 


23 


To  expand  |aiia22«33|  according  to  Cor.  II,  we  have  simply 
to  write  the  elements  for  each  term  with  one  set  of  subscripts 
in  order ;  thus, 

ai(X2<^35   %<^2«3^    «l«2%1    aia2%1    ftl<^2«35   «1«2<^3) 

and  then  for  every  term,  according  as  we  choose  from  columns 
or  rows  in  order,  write  one  permutation  of  the  numbers  1,  2,  3, 
before  or  after  the  subscripts  already  written,  obtaining 

ClllClooCtsS  —  ^'^11^32^^23  "T"  ^31^2^23  —  ^31^22^13  "T"  ^21^%2^''13         <^21^12^33 

or 

%1^22^'33  —  %lC'^23^32  —  <^12<^21^33  ~\~  ^n<^23%l     i     ^13^%  ^32         ^13^^22 '^Sl* 

25.  Theorem.  —  In  any  determinant,  if  the  rows  in  order 
are  made  the  columns  in  order,  the  determinant  is  unchanged. 

The  theorem  is  an  obvious  consequence  of  23  and  24.  The 
following  proof  is  based  directly  upon  the  definition.  Consider 
the  determinants  A  and  A',  which  differ  only  by  making  the 
rows  of  the  one  the  columns  of  the  other.  Every  term  of  A 
contains  an  element  from  each  row  and  column  of  A ;  hence  it 
contains  an  element  from  each  column  and  row  of  A',  and  is 
therefore,  disregarding  the  sign,  also  a  term  of  A'.  Similarly, 
ever}^  term  of  A'  must  be  a  term  of  A.  We  have  now  to  show 
that  the  signs  of  corresponding  terms  are  alike.  Let  the  num- 
bers of  the  rows  and  columns  for  a  term  of  A  be 

a?  y-i  P-)  Ti  0-9  •••  for  the  rows  ; 

r,  t,  a,  s,  771,  ...  for  the  columns. 
Then,  by  hypothesis,  the  numbers  of  the  rows  and  columns  of 
the  corresponding  term  from  A' will  be 

r,  t,  a,  s,  m,  ...  for  the  rows  ; 

«5  y?  i^?  T-i  ^?  •••  for  the  columns. 
The   two   terms   obviously  have   the   same   sign.      Hence  the 
proposition. 


Illustrations : 
a^i  ai2  cii3  ^14 

^21  ^22  ^23  ^24 
%1  ^32  ^33  <^34 
a^i  €1^2  ^43  ^'44 


«U  «21  «31.<*41 
<^12  ^22  %2  ^'42 

ai3  a2s  a^g  a^ 

^14  <^^24  <^34  ^44 


(Xi  bi  Ci  di 

2  ^2     2      *^ 

%  ^3  ^3  ^4 
a^  64  C4  di 


cfci  tta  as  a4 
bi  62  i>3  h 

Ci  C2  C3  C4 

di  do  ds  d^ 


24  THEORY  OF   DETERMINANTS. 

26.  Theorem.  —  In  any  determinant  the  number  of  positive 
terms  equals  the  7iumher  of  negative  terms.  - 

By  23  all  the  terms  of  a  determinant  can  be  formed  by  keep- 
ing the  letters  in  order,  and  making  all  the  possible  permuta- 
tions of  the  subscripts  (or  24,  case  of  the  double  subscripts, 
by  keeping  one  set  in  order  and  permuting  the  other  set) .     AVe 

n\  n\ 

have  to  show,  therefore,  that  -^  permutations  are  even*  and  -^ 

are  odd.*  Let  x  and  y  be  the  number  of  even  and  odd  per- 
mutations respectively;  then  x-\-y  =  n\  If  we  interchange 
any  two  subscripts  in  each  of  the  aj  even  permutations  and  in 
each  of  the  y  odd  permutations,  the  even  permutations  become 
odd  and  the  odd  even.  Since  by  the  interchange  of  two  sub- 
scripts we  could  only  reproduce  permutations  all  different  from 
each  other,  and  already  found  in  the  original  set  of  permuta- 
tions, it  follows  that  x  =  y. 

27.  Theorem.  —  If  two  parallel  lines  (rows  or  columns)  of 
a  determinant  are  interchanged^  the  sign  of  the  determinant  is 
changed,  but  its  numerical  value  is  unchanged. 

Let  A  be  the  given  determinant  and  A'  the  same  determinant 
after  the  A:th  and  rth  rows  have  been  interchanged.  Then 
-A=  A'. 

Let  J^  =  ±  Adj^Bm^C  be  a  term  of  A,  in  which  A,  B,  and  C 
denote  the  product  of  elements  from  all  the  rows  and  columns 
except  the  cZth  column  and  A;th  row,  and  the  mill  column  and 
rth  row.  Then  T  (disregarding  the  sign)  is  also  a  term  of  A', 
for  it  contains  an  element  from  each  row  and  column  of  A'. 
Now  T,  regarded  as  a  term  of  A',  contains  exactly  the  same 
inversions  of  the  columns  as  it  does  when  regarded  as  a  term 
of  A  ;  but  the  number  of  inversions  in  T,  as  to  rows,  when 
considered  as  a  term  of  A',  is  an  odd  number,  more  or  less, 
than  when  considered  as  a  term  of  A.  For,  in  writing  the 
numbers  of  the  rows,   to  determine  the  inversions,   we  write 

*  This  language,  of  course,  signifies  permutations  in  wliicli  the  number 
of  inversions  of  order  is  even  or  odd  respectively. 


GENERAL   PIIOPEKTIES    OF   DETERMINANTS. 


25 


them  just  as  we  would  for  A,  except  that  Aj  and  r  will  have 
changed  places  (dj,  being  found  in  the  rth  row,  and  m^  in  the 
Tcih.  row  of  A') .  Thus  every  term  of  A  is  found  with  the  oppo- 
site sign  in  A',  .•.  —  A  =  A'.  By  25  the  proposition  must  be 
equally  true  for  an  interchange  of  two  columns. 


Illustrations 


ttj  hi  Ci 
ao  69  Co 


«l  ^1  Ci  c?i 
02  &2  Co  d2 
«3  ^3  C3  (k 
a^  64  C4  ^4 


—   [«!  62  C3  +  ^2  63  Ci  +  %  ^1  Co 


f^lXX 


—  ttg  ^2  Ci  —  tto  61  Co  —  a^  63  Co]  =(  7  5 


ai  bi  Ci  f7i 
^2  &2  ^2  c^_> 
a4  64  C4  CZ4 

%  ^3  ^3  ^^3 


ttj  bi  Ci  cli 

«3  ^3  C3  <i3 
a4  &4  C4  ^4 
a2  ^2  ^-2  ^2 


«!  (^1  Cj  bi 


a^  ^4  C4  64 


(ai^gCsfy  =  —  (a^biCsd^)  =  {a^b^Cid^)  =  —  (a2  53C4di)  =  (o3Z>2C4di). 


28.  Cor.  —  If  two  parallel  lines  of  a  determinant  are  iden- 
tical, the  determinant  A^anishes. 

For,  by  the  proposition,  if  the  two  identical  rows  or  columns 
are  interchanged,  the  sign  of  the  determinant  is  changed.  But 
the  interchange  of  two  identical  lines  cannot  affect  the  deter- 
minant.    Therefore 


A=  -A, 
2A  =  0,  or  A  =  0. 


Illustrations 


a  b*c 

=  aec^ 

def 

a  b^^ 

«!  Oj  c 

a.2  b.2  Co  c?2 

ag  60  C2  c?2 

•v 

0-4  64  c 

,^4 

ai 

a.2 

a^ 

«4 

h 

h 

h 

h 

Ci 

C2 

C2 

C4 

^1 

d^d^ 

C^4 

aec  —  dbc  —  a6/=  0. 


0. 


(Ctj  60  Co  C?4)  =  0. 


tti  62  «3  C?4  I  =  0. 


29.    If  in  a  series  of  integers, 

I     /,  a,  d,  c,  Z,  m,  w. 


26 


THEORY   OF  DETERMINANTS. 


the  first  is  passed  over  all  the  others  in  succession  to  become  the 
last,  the  others  remaining  undisturbed,  thus, 

a,  d,  c,  Z,  m,  n,  /, 

the  numbers  are  said  to  have  been  cyclically  interchanged.     It 

is   obvious   that  a  cyclical   permutation  of   n  given   numbers 

can  always  be  effected  by  n  —  1  interchanges  of  two  adjacent 

W       numbers.     Accordingly,   a  permutation  containing  an  odd  or 

{^         even  number  of  inversions  still  contains,  after  a  cyclical  inter- 

^  change,  an  odd  or  even  number  of  inversions  if  n  is  odd ;  if  n 

is  even,  however,   a  pepoiitation  containing  an_o_dd  or  even 

number  of  inversions  will,  after  a  cyclical  interchange,  contain 

an  even  or  odd  number  of  inversions  respectively. 

From  a  given  permutation  of  n  integers  any  other  permuta- 
tion can  be  obtained  by  cyclical  interchanges.     Thus,  from 

faclcegb 
we  get  c  a  gfd  b  e 

as  follows  :  —  cfadegb 

c afd e  g  b 

c a gfd e b 

c  a  g  fd  b  e 
The  groups  in  which  the  cyclical  interchanges  take  place  are, 
of  course,  fade,  fa,  fdeg,  /,  d,  eb. 

30.  The  previous  article  (or  27)  establishes  the  following 
theorem : 

Theorem.  —  If  in  a  determinant  A  any  row  or  column  be 
passed  over  k  rows  or  columns  in  succession,  and  the  resulting 
determinant  be  denoted  by  A',  then 

A=(-1)'=A'. 

Illustrations : 


^iViZiti 

= 

^3  Vs  ^3  ^3 

= 

^3  h  2/3  2=3 

=  — 

x^  ?i  2/1  z, 

x^y^z^h 

XlVlZih 

iCi  fi  yi  Zi 

X2  /a  y-i  Z2 

aJ3  2/3  »3  ^3 

X2y2%2t2 

X2  h  2/2  22 

3^4^4  2/4^4 

x,y,z,t. 

x^y^z^t^ 

x^  U  2/4  z^ 

X'ihy3^3 

^0  2/l  V2lVs\=  —  \  Xi  ^2  VsWo\  =  —\ViX2  2/3  ^^0  |  =  1  «1  2/2  f^3%\' 


GENERAL   PKOPEKTIES   OF   DETERMINANTS.  27 


EXAMPLES. 

1.  The  student  who  has  not  done  the  examples  at  the  end 
of  the  first  chapter  may  attend  to  them  before  proceeding  to 
the  following. 

2.  What  terms  of  |  aib2G^di  \  contain  ftgf^s? 

.  3.    Write  the  terms  of  (a^i  2/2  ^3  2:4  Q  that  contain  ^1 2/4  w^s. 

4.  Show  that  in  a  determinant  of  the  nth  order  only  two 
terms  can  have  (n  —  2)  elements  in  common,  and  that  these 
terms  have  opposite  signs. 

5.  What  is  the  sign-factor  of  the  term  containing  the  ele- 
ments in  the  secondary  diagonal  of  a  determinant  of  the  nth 
order  ? 

6.  Show  that  the  sign  of  a  term  is  independent  of  the 
arrangement  of  the  elements  composing  it. 

7.  Show  that  the  sign  of  a  determinant  is  not  changed  by 
an}^  interchanges  of  rows  and  columns  that  leave  the  same 
elements  in  the  principal  diagonal,  whatever  the  final  arrange- 
ment of  the  elements  in  this  diagonal. 

Syg.  If  a^p  ayy  a^a  •  •  •  ot-m  be  the  final  arrangement  sought, 
a^p  can  be  brought  into  the  first  place  by  2  (^  —  1 )  interchanges 
of  two  rows  and  columns,  etc. 

8.  A  corollary  from  30  is :  Any  element  (X,;^  can  be  trans- 
ferred to  the  first  place  b}'  making  the  ith  row  and  A;th  column 
the  first  row  and  column,  and  then  multiplying  the  determinant 
by  (-1)*+*. 

31.    Theorem. — If  every  element  of  any  line  (row  or  column)     \y 
is  multiplied  by  any  number,  the  determinant  is  multiplied  by 
that  number. 

Since  every  term  of  the  determinant  contains  one  element, 
and  only  one,  from  the  line  mentioned  in  the  theorem,  the  truth 
of  the  proposition  is  evident. 


28 


THEORY   OF   DETERMINANTS. 


Illustrations 


a^r  hyV  CiT 

=  r 

ai  61  Ci 

= 

«!  ftjr  Ci 

=  r2 

«>,, 
^  &1C1 

^2   h   C2 

a2  h  C2 

(12  &2^  ^2 

Clg      63      Cg 

a-g  6g  Cg 

dg  63^  Cg 

<^'2  , 

yhC-I 

«3  , 

7  h  c. 

A  = 


6c  a  (T 
ca  h  b^ 
ah  c  c^ 


then  a&cA  = 


abc  a^  a? 
abc  h^  b^ 
abc  &  <? 


•.  A  = 


1  OL'  a"  . 

1    62  53 

1    C^C^ 


Let  the  student  show  that 


bed  a  a^a? 

=2: 

cda  b  b^  W 

dab  c  c^  c^ 

abc  d  d'  d' 

1  a^  a^  «* 

1     62    53   J4 

1   c2  c^  c^ 
1  d^  d^  d^ 

32.  Cor.  I.  —  Changing  the  signs  of  all  the  elements  of  any 
row  or  column  changes  the  sign  of  the  determinant ;  for  it  is 
equivalent  to  multiplying  the  determinant  by  —  1. 

33.  CoR.  II.  —  If  two  rows  or  two  columns  differ  only  by  a 
constant  factor,  the  determinant  vanishes.  For  we  may  divide 
each  element  by  the  constant  factor,  and  write  this  factor  as  a 
multiplier  before  the  determinant.  Then  the  determinant  van- 
ishes by  28. 


Illustratioi 

IS : 

i 

al  a** 

=  a" 

al  Z 

=0. 

ba    a-+i 

h0L   a 

- 

C  a^  a~  +  2 

eV  a? 

1  5   7 

2  10  6 
345  0) 


1  1  7 

2  2  6 
33  9 


=  0. 


34.  Theorem. — If  each  element  of  any  line  *  of  a  determinant 
is  a  binomial^  the  determinant  equals  the  sum  of  two  determinants; 
the  first  of  which  is  obtained  from  the  given  -determinant  by  sub- 
stituting for  the  binomial  elements  the  first  terms  of  the  binomialSy- 
and  the  second  determinant  is  obtained  from  the  given  determi' 

*  Since  it  has  been  shown  (25)  that  what  is  true  of  the  rows  of  a 
determinant  holds  for  the  columns,  it  will  only  be  necessary  hereafter  to 
state  a  proposition  with  reference  to  either  rows  or  columns. 


GENERAL  PROPERTIES  OF  DETERMINANTS. 


29 


nant  by  substituting  for  the  binomial  elements  the  second  terms  of 
the  binomials. 

By  the  definition  ever}'  term  of  the  determinant  must  contain 
one  of  the  binomial  elements. 

Let  (m-\-n)  b  g  h  k  ,..  I 

be  one  of  the  terms  of  the  given  determinant ;  this  may  be 
written  m  b  g  h  k  ...  l-{-n  b  g  h  k  ...  I. 

Now  the  first  term  of  this  sum  is  a  term  of  the  original  deter- 
minant, with  m  written  for  m-i-n,  and  the  second  term  is  a 
term  of  the  original  determinant,  with  n  written  for  m.-fn.  It 
is  obvious  that  a  similar  statement  applies  to  every  term  of  the 
given  determinant ;  hence  the  proposition. 

Illustrations : 


«!  +  «!    h    Ci 

= 

< 

2l    bi   Ci 

+ 

tti  bi  Ci 

. 

Cl2  +  a2    &2    C2 

a2  h  C2 

0-2    &2    ^2 

a3  +  «3    ^3    C3 

%    h    Cs 

ag     63    C3 

Xi  —  yi  mi  Ui 

= 

Xi  mi  7?i 

— 

2/1  «h  n^ 

0^2-2/2    ^2    ^2 

X2  m-2  «2 

2/2  '^2  no 

a^-2/3     W3     Wg 

Xs  mg  7 

^3 

2/3  'nis 

nsl 

35.  The  preceding  theorem  is  evidently"  capable  of  extension. 
The  same  reasoning  applies  to  a  determinant  any  line  of  which 
is  composed  of  polynomial  elements,  or,  again,  in  which  each 
element  of  every  line  is  a  polynomial.  That  is  to  say  :  If  each 
element  of  any  row  is  a  polynomial  of  q  terms,  each  element  of 
another  row  a  polynomial  of  r  terms,  each  element  of  another 
row  a  polynomial  of  s  terms,  etc.,  the  given  determiyiant  is  the 
sum  of  sxq  X  r...  determinants.     Thus  : 


a-\-b  -\-  c    m  —  n 
d+e-f     o+p 
g  —  h-\-k     q  —  r 

t 
u 

V 

1 

i 

7 

m 
0 
Q 

—  n    t 
4-i)    u 

—  r    V 

-f- 

b    m  —  n 

e     0  -\-p 

-h     q  —  r 

t 

u 

V 

c    m  —  n    t 

—f     o+p    u 

k     q  —  r     V 

— 

a  m  t 
d    0    u 
g    q    V 

+ 

a  —  n    t 
d     p    u 
g  —  r    V 

+ 

b    m    t 

e    0    u 

—  hqv 

b  —  n    t   + 
e     p    u 
—  h  —  r    V 

A 

1  n 
0 

i  t 

u 

V 

H 

h 

c  —  n 

-f    P 
k-r 

t 
u 

V 

30 


THEORY   OF   DETERMINANTS. 


36.  Reciprocally,  If  q  determinants  differ  from  each  other 
only  in  a  single  line,  the  sum  of  these  determinants  is  a  single 
determinant,  derived  from  any  of  the  given  determinants  by  sub- 
stituting for  the  elements  of  the  line  which  is  different  in  each 
of  the  q  determinants,  the  sum  of  the  corresponding  elements  of 
the  q  determinants. 

Illustrations  : 


ab  c 

+ 

ab  c 

+ 

mn  0 

= 

a                  b                 c        \ 

def 

X  y  z 

ab  c 

d-{-x  —  m    e-\-y  —  n   f-\-z  —  o 

ghk 

g  hk 

ghk 

g                  h                 k 

The  student  may  show  that 

«!     «i     61     Ci 

— 

tti   bi  Ci   0 

=  0. 

0-2     «'2     h     C2 

a^   62   C2   0 

0         ttg      bs      C3 

^3   ^3  C3  ag 

0 

a^   &4 

C4 

a^   b^   C4   a^ 

37.  Theorem.  —  A  determinant  remains  unchanged  if  the 
elements  of  any  line  be  increased  or  diminished  by  equal  multiples 
of  the  corresponding  elements  of  ayiy  parallel  line. 

We  are  to  show  that 


tti    61    Ci 

tta  62  ^2 
^3  ^3  ^3 


«i  ^1  Ci±qiai±q2bi±...±li 
a2  62  C2±qia2±q2b2±...±L 
<X3   &3   Cs±qias±q2b3±...±l 


d,  ...  h 

4  •••  ^s 


a„  6„  c,,±qia^±q2b^±...±l,  d,  ...  /, 


Calling  the  first  determinant  A,  and  the  second  A',  we  have,  35, 
A'  = 


a,  bi  Ci  d,  . 

(/2     60     Cg     (^2    • 

±qi 

± 

«i  ^1  «i  ^^1  • 

0^2    ^2    «2    C^2  • 

./2 

aa  62  ?2   ^2  • 

(^n    ^n  ^n    <^n    • 

ai  61   61  di  . 

a2    ?>2    ^2    ^^2   • 

»n  K    «n   <^n^- 

«n    K    K    dn    . 

.  L 

«n   ^n    ^«    dn    . 

..    I, 

:t<?2 


Whence,  since  all  the  determinants  of  this  series,  except  ILe 
first,  A^anish,  A  =  A'. 


GENERAL  PROPERTIES  OF  DETERMINANTS. 


31 


This  theorem  is  of  great  importance  in  simplifying  and  ex- 
panding determinants.     Thus : 


lab-\-c  =  la 
1  b  c  +  a  1  b 
1    c   a  +  b         1    c 

a  a+3 a+6 
a  +  1  a  +  4  a-\-7 
a  +  2  a  +  5  a-f-8 


a  +  6  +  c  |  =  (a4-6  +  c) 
a  +  6  +c 


1  a  1 
1  &  1 
1  c  1 


=  13  (a 


1)  3  (a +  4)   3  (a +  7) 

1  a  +  4         a-f7 

2  a-l-5         a  +  8 


=  0, 


The  second  determinant  is  obtained  by  adding  the  second 
and  third  rows  to  the  first  row.  7 

=  -16. 


1 

1 

1 

1 

= 

-11 1 1 

1- 

-1 

1 

1 

/0%2  2 

1 

1 

-1 

1 

//O  2X2 

1 

1 

1 

-1 

J  02  2  0 

=r — 

0  2  2 

=  —8 

0  1  1 

202 

1  0  1 

2  20 

1  1  0 

The  second  determinant  is  obtained  by  adding  the  first  row  to 
each  of  the  others. 

The  third  determinant  is  obtained  from  the  second  by  ob- 
serving that  as  all  the  elements,  except  one,  of  the  first  column 
of  that  determinant  are  zeros,  all  the  terms  vanish  that  do  not 
contain  (—1). 


7 

13 
3 


11 
15 
9 


4 
10 
6 


7 

11 

13 

15 

1 

3 

-240. 

4 

=  3 

7 

-10 

-10 

=  3 

10   10 

10 

13 

-24 

-16 

24   16 

2 

1 

0 

0 

=  30(16-24)  = 

The  third  determinant  is  obtained  from  the  second  by  subtract- 
ing three  times  the  first  column  from  the  second  column,  and 
twice  the  first  column  from  the  third. 


/ 


Minor  Determinants. 

38.  If  in  a  determinant  any  number  of  rows  and  the  same 
number  of  columns  are  suppressed,  the  determinant  consisting 
of  the  remaining  elements  (their  relative  positions  being  undis- 
turbed) is  called  a  minor  of  the  given  determinant. 

If  one  row  and  one  column  are  suppressed,  the  result  is  a 
principal  minor,  or  a  Jirst  minor ;  if  two  rows  and  two  columns 


32  THEORY   OF   DETERMINAMTS. 

have  been  erased,  a  second  minor ;  and  so  on.  The  elements 
common  to  the  suppressed  rows  and  columns  also  form  a  deter- 
minant called  the  complementary  of  the  minor,  formed  from  the 
rows  and  columns  that  were  left  undisturbed  in  the  original 
determinant. 


Thus, 


«3    ^3 


and 


C4    ^4 


are    complementary   minors   of 


I  «!  62  C3  c?4  I ;  also  I  «i  62 1  and  |  Cg  (^4 1,  or  &2  and  |  a^  C3  CZ4 1 ,  are 
complementary  minors  of  |  aj  62  C3  CZ4 1 .  |  c^  d^  \  and  |  a.,  64  e^  \  are 
complementary  minors  of  \ai'b2C.id^e^\.  In  general,  if  the 
determinant  is  of  the  nth  order,  two  complementar}"  minors  will 
be  of  the  rth  and  {n  —  r)th  orders  respectively.    A  determinant 

n^  (71  ~  IV 
of  the  71  th  order  has  rr  first  minors,  — ^^ L  second  minors, 

etc. 

Since  we  usually  denote  a  determinant  by  A,  it  is  convenient 
to  denote  the  minor  obtained  by  suppressing  the  row  and 
column  of  a^  by  A^g ;  that  obtained  by  suppressing  the  row 
and  column  of  d^  by  A^^^,  etc. 

Similarly,  a  second  minor,  obtained  by  suppressing  the  rows 
and  columns  of  b^  and  c^,  is  denoted  by  Abh.Cr ;  and  so  on. 

Equally  efficient  notations  are  :  D  )       and  D}     ^^  -.    for  the 

minor  obtained  by  suppressing  the  Zth  row  and  mth  column, 
and  the  minor  obtained  by  suppressing  the  Zth,  wth,  and  tih. 
rows,  the  mth,  rth,  and  h\h  columns  respectively  of  |  %,i  | . 

39.  Since,  by  definition,  every  term  of  a  determinant  con- 
tains one,  and  onl^^  one  element  from  any  line,  the  determinant 
must  be  a  linear  homogenous  function  of  the  elements  of  any 
one  row  or  column.     Thus  : 

I  tti  62  C3 ...  ^^  1  =  «!  J.1  +  a2-42  +  a^A^  -\ f- a,,A,, 

=  «!  A  +  \B,  +  ci  Ci  +  •••  +  /i  A 

=  ci  Ci  +  C2  a,  +  C3  (73  + .-.  +  c„  a 

in  which  ^1,  A2...A,, ;  Ci,  (72...C„  ;  etc.,  denote  functions  of  the 
elements  found  in  the  rows  and  columns  outside  of  the  particu- 


GENERAL  PEOPERTTES  OF  DETERMINANTS.     33 

lar  line,  in  terms  of  which  the  development  is  given.  In  the 
next  article  we  shall  find  the  values  of  these  functions.  Since 
we  may  regard  the  determinant  as  a  function  of  r^  independent 
quantities,  each  of  the  coefficients  A^^  A^^  ...,  may  be  obtained 
by  differentiating  |  aiftgCs-.-Z^  |  with  reference  to  the  quantity 
whose  coefficient  is  desired.  Introducing  this  concept,  the 
equations  above  written  become  respectively 

I  ai62C3...?„ 


,  cZA   ,       cZA   ,       cZA  , 
cZttj          da^         da^ 

cZA 

••  +  «,-— 

da^ 

cZA   ,   ,   cZA  ,       cZA   , 
dai         "Oi         dci 

J  dA 

^  d\   ,       clA  ^      cZA   , 
dci          cZcg         (Zc3 

This  notation  is  often  employed. 

/^^^s^xI'heorem.  —  The  coefficient  of  any  element  in  the  expan- 
sion of  a  determinant  is  the  first  minor  obtained  by  suppressing 
the  row  and  column  to  which  the  elemeyit  belongs.  This  7riinor 
is  taken  with  the  +  sign,  if  the  sum  of  the  row  and  column 
numbers,  to  which  the  element  belongs,  is  even;  if  this  sum  is 
odd,  the  minor  has  the  —  sign. 

Consider  the  determinant  A  =  2  ±  «i  &2  ^3 . . .  In-,  and  suppose 
A  to  be  written, 

A  =  ai^1l  +  «2^2  +  «3^3+    •••    +«nA-  (1) 

We  can  collect  all  the  terms  of  A  that  contain  ai,  and  write 
this  element  as  a  factor  of  the  polynomial  that  results  ;  we  can 
do  the  same  for  a^',  a^,  and  so  on,  for  each  element  of  the  first 
column.  These  pol^'nomials  are  ^1,  A2,  A^,  etc.  Now  A^  must 
be  composed  of  all  the  terms  of  A  that  contain  no  elements 
from  the  first  row  or  column ;  hence  A^  can  be  obtained  from 
2  ±  «!  &2  C3 ...  Z^  by  considering  ai  as  fixed,  and  making  all  the 
possible  permutations  of  the  subscripts  of  the  remaining  letters, 
i.e.,  by  multiplying  aj  by  IS  ±  &2  ^3  •••  hv 

Hence,  ^1  =  A„j,  the  minor. obtained  by  suppressing  the  first 
row  and  first  column  of  A. 


34 


THEORY   OF   DETERMINANTS. 


V' 


Now,  we  can  bring  0^2  into  the  first  place  by  one  interchange 
of  rows  :  we  then  have  A  =  —  2  ±  ag  61  Cg  ...  Z„.  Employing  the 
same  reasoning  as  before,  A2  must  be  obtained  by  multiplying 
^2  by  —  5  ±  &i  Cg ...  Z„  ;  whence  A2  =  —  Aa^.  Again,  a^  can  be 
brought  into  the  first  place  by  two  interchanges  of  two  rows  ; 
whence  A^  =  A^g,  and  so  on.  Finally,  a^  can  be  brought  into 
the  first  place  by  n  —  1  interchanges  of  two  rows ;  hence,  as 
before,  A  =(-!)""' Aa«. 

Substituting  these  values  of  Ai,  Az,  etc.,  in  (1), 

A  =  tti Aa^  -  a2^a,  +  %Aa3 h  ( - 1 ) ""^ a^ Aa«. 

Since  the  columns  may  be  made  rows,  it  is  evident  that 
A  =  aiAa,-&iA6^+CiAc, +  (-l)"-*ZiAv 

It  remains  to  be  shown  that  the  proposition  holds  for  an  ele- 
ment not  in  the  first  row  or  column,  that  is,  A^^  =  (  —  1)*'^''  Aaa, 
for  the  coefficient  of  the  element  in  the  iih  row  and  Jcth.  column. 
We  may  transfer  the  ith  row  to  the  first  place  by  i—  1  inter- 
changes of  two  rows,  and  the  Jcth  column  may  likewise  be  made 
the  first  by  A:  —  1  interchanges  of  two  columns.  The  element 
under  consideration  is  now  in  the  first  place.  Calling  the  trans- 
formed determinant  A',  we  have 

A  =  (-l)*+*-2A',  orA  =  (-l)^+*A'. 
Whence      ^,,  =  (  - 1  ^ +*  Aa^, . 

41.  CoR.  I.  — A  determinant  can  be  developed  in  terms  of 
the  elements  of  any  line  and  their  principal  minors.  The  signs 
are  alternately  +  and  —  ;  and  the  fii'st  term  is  +  or  — ,  accord- 
ing as  the  number  of  the  line  is  odd  or  even. 


Illustrations : 

tti  bi  Ci  f?i 

=  «! 

62  C2  d2 

-a^ 

h  Ci  c^i  -\-  as 

b,  c,  d, 

-a^ 

61  Ci  dy 

O2  ^2  ^2  d.2 

h    Cg    dg 

h  Cg  dg 

h  C2  di 

62  C2  (?2 

«3  ^3  Cg  ^3 

64  C4  d^ 

64   C4  ^4 

64  C4  d4 

h<hd^ 

^4  64  C4   6^4 

=  —  61 1  as  Cg  ^4  I  4-&2  1  «i  Cg  d^  I  — ftglai  Cg  d^  1  -f  ?>4  1  «l  C2  dgj 

=      Ci I a2  63  (^4 1  — C2 1  ai  &3  ^4 1 4-C3|ai  &2 1?4 1  — C4  | «i  ^2  ^3 1 

=  -  «4|  ^  C2  (^3  I  +^4  1  %  C2  C?3  I  -C4|ai  hi  ^3  |  -f-Cf4|  tti  &2  C3I 


GENERAL  PROPERTIES  OF  DETERMINANTS. 


35 


42.  41  obviously  gives  a  ready  way  of  expanding  any  deter- 
minant.* For  we  may  express  the  given  determinant  in  terms 
of  the  elements  of  any  line  and  their  principal  minors  ;  these 
minors  will  be  determinants  of  the  (n  —  1 )  th  order.  By  a 
second  application  of  41,  each  of  the  minors  in  the  first  expan- 
sion ma^'  be  expressed  in  terms  of  the  elements  of  any  line  and 
their  principal  minors,  which  minors  will  be  of  the  (?i  — 2)th 
order.  So  by  successive  application  of  41,  an}'  determinant 
may  be  expressed  in  terms  of  determinants  of  the  second  order  ; 
and  these  latter,  being  binomials,  can  be  at  once  written  out. 
Thus : 


1    2    3 

=  13    4 

-2 

2    3+3 

2    3 

2    3    4 

4    5 

4    5 

3    4 

3    4    5 

1(15-16) -2(10-12) +3(8-9)  =0. 

\aib2Csd^\^=ai\b2Csd^\—a2\biCsdi\-^as\biC2di\—a^\biC2ds\ 

=  ai[b2\csd^\-b^\c2d^\+bi\c2ds\2-a2lbi\Csd^\-b^\cidi\-\-bi\cids\2 

+asibi\c2di\—b2\cid^\-\-b^\cid2\']—a^lb^\c2ds\—b2\cid-^\-\-b.i\cid2\^ 

=  aiboC^d^  —  aj  62^4^3  —  «i  6302(^4  +  ai  6304^2  +  aibiC2ds  —  a^b^c^di 

—  a2  &i  C3  C?4  +  (^2  ^1 C4  c?3  + 

-\-a^biC2d^  —  a.;ib^G^d2  — 

—  a^b^Cid^  +  0146103^2  +  aib2Cid^  —  a^biC^d^  —  a^b^Cid2  +  aJy^Cidi 

43.  As  another  corollary  from  40,  it  is  evident  that  if  all 
the  elements  of  any  row  of  a  determinant  except  one  are 
zeros,  the  determinant  equals  this  element  into  its  corresponding 
minor,  taken  with  the  proper  sign.  Thus,  if  the  element  is  in 
the  ith  row  and  A:th  column,  i.e.,  a,.;^,  then  A  =  (  — l)'+*a,4Aa,.^. 

Illustrations  : 

0-2 
1  1 
0     1 


5  643 

=  -2 

6  4  3 
1  1  1 
1  2  1 

=  -2 

2-0  0  0 
3  111 

3 

=  2 

-2 

-3 

1 
0 

1 

0 

*  Compare  15. 


t  Compare  6. 


36 


THEORY   OF  DETERMINANTS. 


The  student  may  establish  the  following : 


«! 

h 

c,d. 

0 

h 

c,d. 

0 

0 

CsCh 

0 

0 

0  d. 

ctiboCgd^. 


0  0  0  di 

0   0  Ca  ^2 
0   &3  Cg  d^ 
. — ^^b^c^d^ 


a^hc^di. 


44. 


From  the  last  two  examples  it  appears  that  if  all  the 
elements  on  one  side  of  either  diagonal  are  zeros,  the  determinant 
reduces  to  a  single  term,  viz.,  the  term  composed  of  the  elements 
in  the  diagonal  which  contains  no  zero  elements. 


2       1  - 

-4  -3 
6      5- 

7 
8 
9 

=  2 

1  1 

2  3 

3  5 

7 
8 
9 

• 

Show  that 

0 

a 

0 

.      /3 
J'    (3' 

'I" 

= 

1 
al3y 

EXAMPLES. 


1.    Show  that  the  following  determinant  vanishes  : 


2. 


1  1  1 

a'  (By  /S'  ya  y'  a/B 
a"fty  /S'V"  y"^^ 
This  can  be  readily  established  by  multiplying  the  columns 
by  (Sy,  ya,  a^,  respectively,  and  then  dividing  the  first  row  by 
aySy.  A  similar  reduction  can  he  effected,  in  general,  tchenever  it 
is  desired  to  reduce  a  determinant  to  one  in  which  the  elements 
of  one  line  are  units. 


4.   Find  the  expansion  of  A 


4  2  5  10 
116  3 
73  0  5 
025  8 


We  notice  that  20  is  the  L.C.M.  of  the  elements  in  the  first 
row;  hence,  multiplying  the  columns  in  order  by  5,  10,  4,  2, 
there  results 


A  = 


1 


5.10.4-2 


20  20  20  20 

5   10  24  6 

— 

35  30  0   10 

0  20  20  16 

1 

1 

1 

11 

5 

10  24  6  1 

7 

6 

0 

2 

0 

5 

5 

4 

GENERAL  PEOPERTIES  OF  DETERMINANTS. 


37 


/ 


Now,  subtracting  four  times  the  first  row  from  the  fourth 
row,  two  times  ,the  first  row  from  the  third  row,  and  six  times 
the  first  row  from  the  second  row,  the  last  determinant  becomes 


1  1 

1    1 



_ 

-1  4 

18 

. 1 

71  -14     18 

1   4 

18  0 

5  4 

-2 

-3        6    -2 

5  4 

-2    0 

-4  1 

1 

0        0       1 

4   1 

1    0 

=  —6 

71 

—  7 

=  -6 

64  -7   =-3 

-1 

1 

0       1 

5. 


also 


(Xj  (X2  (X3 
61  &2  &3 
C.i   C2   C3 


1  a  a' 
IIS  13' 


1 

1                     1                     1 

aia20 

C2    C3 

^3      * 
< 

^2  (Xs  &i       «!  0-3  62       ^1  Cf' 

0  1              1 

1  ttg  62  Cj     a2  ^3  ^1 
1     asbiC2    a2^C3 

2&3 
2C3 

=  (/3-7)   (y-a)   (a-;8) 


A  vanishes  if  a  =  /S,  or  j8  =  y,  or  a  =  y  ;  hence  a  — ^,  /?  —  y, 
and  a  —  y  must  be  f aotors  of  A.  Now  the  product  of  the  three 
differences  is  a  function  of  the  third  degree  in  a,  ^,  y ;  so  is  A  ; 
hence  the  product  of  the  three  differences  can  differ  from  A  only 
by  a  constant  factor.  Comparing  the  term  j3y^  (the  principal 
term),  we  see  the  factor  mentioned  is  +  1. 

7.    Show  that 
1111    =_(^_^)(,_8)(^_„)(^_S)(,_^)(^_8). 

a  /3  y  8 
a'  /32  y2  §2 
a^   ^3    ^3    33 

Notice  that  Examples  6  and  7  give  in  determinant  form  the 
product  of  the  differences  of  the  roots  of  an  equation  whose 
roots  are  a,  ^,  y,  ... 


8.    Expand 


8  7    2 

20 

;  also 

3  1    4 

7 

5  0  11 

0 

8  10 

6 

1        a 

-a        1 
(S   -y 


7 

1 


38 


THEORY   OF   DETERMINANTS. 


Expand  the  first  determinant  in  terms  of  the  elements  of  the 
third  row  and  their  principal  minors,  since  two  of  these  elements 
are  zero ;  then  observe  that  two  elements  in  a  row  of  each  of 
the  resulting  determinants  are  unity ;  hence,  each  determinant 
can  be  readily  reduced  to  one  of  the  next  lower  order  by  35 
and  42.  «^  >>  + 

:  V  d^  +  b^e^  +  c'f-2  bcefr-  2  cafd  -  2  abde. 


c      d 

;  also 

d     c 

a      b 

b  -a 

P 

y, 

y' 

-P' 

1 

a' 

a' 

1 

1        o 
-a        1 

-y     P 

11.    Establish    the    following   identity,    and   express   either 
determinant  as  the  product  of  four  linear  factors  : 

0  X  y  z 

X  0  z  y 

y  z  0  X 

z  y  X  0 


0  111 

= 

1  Oz'f 

1  z'O  x" 

1  y^z'O 

12.    Simplify 


13.    Show  that 


ai-\-hi  +  ki    a2  +  ?h-hh    cis-^h  +  h     1 

Ci-j-Jh-\-h    C2-{-hs-\-h    Cs-hh-{-h     1 
1  1  10 


X  y-^z  +  t 

y  z-\-t-{-x 

z  t  +  x-^y 

t  x-\-y-{-z 


x-j-y  z  +  t 

y-\-z  t  +  x 

z-{-t  x  +  y 

t  +  x  y-\-z 


=  0. 


14.  Express  as  a  single  determinant: 

(1)  \a^b^Cs\-^\a2b^c,\-\a^biC,\. 

(2)  I  Qq  62  Cs  I  —  I  ao  63  Cs  I  —  I  ai  63  Cj I  + 1  tti  62  C5I . 

15.  (Xi+aa+tta    (i2-\-^h-\-(^A    «3+«4H-«i    a4+^i+«2 
6i+?>2+?>3     b2+h+b^     ^3+^+^     64+^+^2 

Cl+C2-f-C3       C24-C3+C4       C34-C4+C1       C4+C14-C2 
d^-\-d2  +  d^      C?2  +  <^3  +  <^4      C^3  +  ^44-f?l      C?4-f(Zi  +  d2 

=  3  1  ciibiC^d^  |. 


GENERAL  PROPERTIES    OF   DETERMINANTS. 


39 


16. 


17. 


sin^^ 


6sin^       1  cos^ 

csin^     cos^       1 


cv'-{h^+c'-2bGGo^A). 


a+h-\-nG  (n  — l)a  {n  —  \)h 
(n  — l)c  h-\-c-\-na  (ti— 1)6 
(w  — l)c       (n— l)a      c-\-a-\-nh 


18.  {a+hy  d"  (?         =2abc(a-{-b-hcy. 

a'  {b-^cy  a' 

b'  b^   '      {c^ay 

19.  What  is  the  coefficient  of  ^34  in  |ai5 1? 

20.  From  the  first  five  rows  of  {dib^c^d^e^fQ  {/y/ig]  write  all 
the  possible  minors  that  can  be  formed,  and  their  complemen- 
taries. 

How  many  minors,  each  a  determinant  of  the  hth.  order,  can 
be  formed  from  any  k  rows  of  |  ai„  |  ? 

21.  If  each  of  the  elements  of  any  line  is  the  sum  of  the  cor- 
responding elements  of  two  or  more  parallel  lines,  multiplied 
respectively  by  constant  factors,  the  determinant  vanishes. 

22.  Show  that 


tti    61    Ci 

= 

10    0    0 

= 

«i  h  Ci  2/1 

= 

«1    61    Ci   Ui   Vi    . 

«2    \    ^2 

Xi    «!    61    Ci 

^2  h  C2  2/2 

a.2    &2    ^'2    ^^2    '^2 

«3    ^3    C3 

072    ^2    ^2    ^2 

«3    h    C3    2/3 

ttg     ?>3    C3     W3    -^3 

x^  as  h  C3 

0    0    0    1 

0    0    0   1    V4 

0    0    0   0    1 

From  this  example  it  appears  that  any  determinant  may  be 
expressed  as  a  determinant  of  higher  order  by  tvriting  a  zero 
above  every  column,  prefixing  a  1  to  the  row  of  zeros  thus  formed, 
and  filling  in  the  new  column  having  1  at  the  top  with  any  n 
finite  quantities. 

,K  23.  If  in  any  determinant  each  element  of  the  first  row  is 
unity,  and  if  each  element  of  every  other  row  is  the  sum  of  the 
elements  above  and  to  the  left  of  it  in  the  preceding  row, 
commencing  with  the  element  directly  above,  the  determinant 
equals  1. 


40 


THEORY   OF   DETERMINANTS. 


24.  Any  determinant  of  order  w,  in  which  one  element  is 
zero,  is  equal  to  the  product  of  two  factors,  one  oC  which  is  a 
determinant  of  the  nth  order,  in  which  every  other  element  of 
the  row  and  column  containing  the  zero  is  unity. 

25.  If  in  any  determinant  the  first  element  is  zero,  and  if 
each  of  the  remaining  elements  in  the  first  row  and  first  column 
is  unity,  the  determinant  is  unchanged  when  each  element  of 
the  minor  corresponding  to  the  zero  element  is  increased  or 
diminished  by  the  same  quantity. 

26.  A  determinant  of  the  nth.  order  is  expressible  as  the 
sum  of  n  determinants,  the  first  of  which  is  obtained  by  chang- 
ing into  zero  each  element  of  any  line  except  the  first  element, 
the  second  by  changing  into  zero  the  elements  of  the  same  line 
except  the  second  element,  and  so  on. 

27.  If  in  two  determinants  A,  A'  of  the  nth  order,  the  first 
row  of  A  is  the  last  row  of  A',  the  second  row  of  A  the  (n  — l)th 
row  of  A',  the  third  row  of  A  the  (n— 2)th  row  of  A',  and  so 


on 


then 


A=(-l) 


n  (n-1) 


2      A'. 

28.  If  in  two  determinants  A,  A'  of  the  nth  order,  the  first 
row  of  A  when  reversed  is  the  last  row  of  A',  the  second  row 
of  A  when  reversed  is  the  (n—l)th  row  of  A',  the  third  row  of 
A  when  reversed  is  the  (w  — 2)th  row  of  A',  etc.  ;  then  A  =  A'. 

45.  Theokem.  —  If  the  elements  of  any  line  in  a  determinant 
are  respectively  multiplied  by  the  complementary  minors  taken 
alternately  plus  and  minus  {i.e.,  the  co-factors)  of  the  correspond- 
ing elements  of  any  parallel  line,  the  sum  of  the  products  is  zero. 

Consider  the  two  determinants, 
A 


oti  bi  Ci 

02    &2    C2 

...     I, 
...     k 

and 

A'  = 

ai  bi  Ci  ... 
^2  Z>2  C2   ... 

O'lc  b„  Cj, 

...    Ifi 

a,  6,  c,  ... 

Ik 

%  K  % 

...    Ip 

«A  h  Ck  ••• 

''k 

ttn    K   C„ 

...    t^ 

CtnKCn    ... 

"rt 

GENERAL  PROPERTIES  OF  DETERMINANTS.     41 

where  A'  differs  from  A  only  in  having  the  Jcth  and  pth  rows 
identical.  Employing  the  notation  of  39,  and  expanding  in 
terms  of  the  elements  of  the  pth  row, 

Comparing  these  two  expansions,  we  observe  that  the  second 
may  be  obtained  from  the  first  by  substituting  for  the  elements 
of  the  pth.  row  of  A  the  elements  of  the  Zcth  row ;  that  is  to 
say,  if  the  elements  of  the  kth  row  of  A  are  multiplied  by  the 
co-factors  of  the  corresponding  elements  of  the  pth  row,  the 
result  is  A' ;  since  A'  =  0,  the  proposition  is  established. 

Illustrations : 

If  in  (aib2Cs)  =  ai  (b2C.])  —  a2  (biC^) -{- a^  (biC2)  we  multiply 
the  elements  of  the  second  column  respectively  by  the  com- 
plementary minors  of  «!,  ag,  as,  there  results 

&i  (&2C3  —  bsC2)  —  62  (^1^3  —  &3C1)  +  &3  (^1^2  —  &2C1)  =  0. 
Let  the  student  prove  the  proposition,  using  a  determinant 
of  the  fourth  order. 

46.  A  determinant  is  said  to  be  zero-axial  if  each  element  of 
the  principal  diagonal  is  zero.  Thus  the  following  are  zero- 
axial  determinants : 


0 

\ 

Ci 

» 

0    61  Ci  c?i 

a2 

0 

C2 

(X2    0     C2    C?2 

as 

h 

0 

04    ^4    C4    0 

47.  Theorem.  —  Any  determinant  may  be  decomposed  into  a 
sum  of  zero-axial  determinants :  the  first  of  these  is  obtained  by 
substituting  zero  for  each  element  of  the  principal  diagonal  of  the 
given  determinant ;  the  next  ?i,  by  miiltiplying  each  element  of 
the  principal  diagonal  by  its  complementary  minor  made  zero- 

n 
axial;  the  next  -(71— 1),  by  multiplying  each  product  of  pairs 

of  elements  of  the  principal  diagonal  by  its  complementary  minor 
made  zero-axial^  and  so  on. 


42 


THEORY   OF   DETERMINANTS. 


In  A^"' =  \a1b2C3...  l,,\   change  the  elements  of  the  principal 

denote  the  resulting  determi- 

h 


diagonal  into  zeros,  and  let  aJ" 
nant.     Whence, 

0    bi  Ci 


Ar^ 


a^  0 
ttg  bs 


b^  c„  ...  0 


Let  aJ"""^^  denote  the  minor  of  Aq"^  obtained  by  suppressing 
any  one  row  of  Aq"  ;  Aq""  denote  the  minor  obtained  by  sup-- 
pressing  any  two  rows  of  Aq'^  ;  and,  in  general,  let  Aq*'"'^  denote 
the  minor  obtained  by  suppressing  any  i  rows  of  Ao"\  Also 
let  C2  denote  any  product  of  the  elements  of  the  principal 
diagonal  of  A^"^  taken  2  and  2  ;  Cg  any  product  of  those  elements 
taken  3  and  3  ;  and,  in  general,  (7,  any  product  of  the  elements 
of  the  principal  diagonal  of  A^**^  taken  i  and  i.  Now,  Aq** 
evidently  contains  all  those  terms  of  A^*"^  which  involve  no 
element  from  the  principal  diagonal.  CiAo"~  must  be  one  of 
those  terms  of  the  series  which  involve  only  a  single  element 
from  the  principal  diagonal  of  A^**^ ;  consequently  :S  Cj  Ao**"^^ 
will  be  the  sum  of  all  the  terms  that  contain  only  one  element 
from  the  principal  diagonal  of  A^''^  Similarly,  1<C2'^1'^~^^  will 
be  the  sum  of  all  the  terms  that  contain  only  two  of  those 
elements.  And,  in  general,  SCfAj""'^  will  be  the  sum  of  all 
the  terms  containing  i  elements  of  the  principal  diagonal  of 
^(n)^     Whence, 


(n-2) 


A("> = Ar + :s  o,  Ar''+ 2  c,  Ar""+ :s  c.a^o 


(n-3) 


+  ...+2C,A 


(n-i) 


It  is  to  be  noticed  that  Aq^'  =  0  ;  i.e.,  there  is  a  break  in  the 
series,  —  there  being  no  term  containing  only  7i  —  l  of  the 
elements  in  the  principal  diagonal. 


Illustration 


«i  2/1  2^1 


^2  2/2  ^2 
^3  2/3  ^3 


0   2/1  ^1 

X2  0    Z2 

^3    2/3    0 


+  Xi 


0  Z2 

2/3  0 


+2/2 1 0  % 


+  z, 


0  2/i|+«i2^2^3- 
X2O  I 


GENERAL  PROPERTIES  OF  DETERMINANTS. 


43 


48.  Theorem.  —  If  each  consecutive  pair  of  elements  in  the  first 
row  of  a  determinant  A  is  taken  with  each  pair  of  corresponding 
elements  of  the  other  consecutive  rows  to  form  determinants  of 
the  second  degree^  and  if  these  determinants  of  the  second  degree 
are  used  in  order  as  the  elements  of  a  new  determinant  A',  then 
A  equals  A'  divided  by  the  product  of  all  the  elements  except  the 
first  ayid  last  in  the  first  row  of  A. 

We  are  to  show  that 


1 


ai  6i  Ci 

ttg    &2    ^2 
«3    ^3    C3 

...  ri  Zi 
...  r2  k 

...    7*3    k 

ttn   K  Cn 

...  r.  In 

h^c^d^...r^ 


ai  b, 

aa  &2 

62  C2. 

... 

ri  Zi 
^2  Z2 

«3  h 

&1    Ci 

^3  C3 

... 

^3/3 

ai  61 

&1    Ci 

... 

Tn   In 

calling  the  first  determinant  A,  and  the  second  A'.  Multiplying 
the  first  column  of  A  by  —  61,  and  the  second  column  by  %, 
and  adding,  there  results 


-6iA  = 


0 
a^  bi  -j-  cii  62 
a^bi  +  a^b^ 


Ci  ... 

C2  ... 


Now,   multiplying  the  second  column  by  —  Ci,   and  the  third 
column  by  &i,  and  adding,  we  have 


61C1A 


0  0 

a2bi  +  aib2     —  &2Ci  +  &iC2 
a^bi-^a^bs     —63^1  +  ^1^3 


^3    h 


—a^bi-\-a^bn     -b^c^  +  b^c^     c^...r^    l„ 
Proceeding   in   a   similar    manner,   we   have,    after    (n—l) 
transformations, 

{-ir-'b,c,d,...kA 

0  0  0         ...  0  Zi 

—  a2bi  +  aib2    —b2Ci-\-biC2    —  CgcZi +  01^3 ••• —^2^1  + ^1^2    k 

—  a^bi-haibs    —  ^s^i  +  ^Cg    —  Cgdj  H-CicZg...  — rgZi  +  riZg    Z3 

-a„6i+ai6„    -6„Ci-f&iC„    -c^di+CicZ„... -r„Zi  +  riZ„  Z« 


44 


THEORY   OF   DETERMINANTS. 


Now,   applying  43,   and   dividing  by   (  — l)**~^6iCicZi  ...  ^i, 
we  have  ^^ 


|«i  &2I  l&i  C2 
l«i  &3I  1^1  C3 


7i  ^3 


-f-  6iCicZi...ri,^ 


«!  &n|  1^1  c„|  ...  |ri  Z, 
which  establishes  the  x>roposition. 

49.  Since  by  the  preceding  proposition  any  determinant  of 
the  njih  order  may  be  reduced  to  one  of  the  (n— l)th  order,  we 
have  another  means  of  simplifying  any  given  determinant. 
The  proposition  is  especially  advantageous  in  the  reduction  of 
determinants  whose  elements  are  given  numbers.  Thus  : 
A  = 


12  3  4 

_1 

3  2  14 

6 

13  4  5 

5  4  3  2 

-4   -4 

8 

=  -4 

-1   -1 

2 

1   -1 

-1 

1   -1 

-1 

-6   -6 

-6 

1       1 

1 

24. 


Here  we  can  mentally  reduce  the  determinants  of  the  second 
order  obtained  by  combining  the  first  pair  of  elements  of  the 
first  row  with  the  corresponding  elements  of  the  other  rows, 
and  obtain  the  elements  of  the  first  column  of  the  new  deter- 
minant, thus :  lx2-3x2  =  -4;  1x3-1x2  =  1;  1x4 
—  5x2  =  — 6.  For  the  elements  of  the  second  column  we 
have  similarly :  2x1-2x3=  —  4;  2x4-3x3  =  —  1; 
2x3  —  4x3  =  —  6;  and  so  on. 

Let  the  student  apply  the  proposition  to  show  that 


1      1 

1  \-\-x 
1      1 
1      1 


1 

1 

1 

1 

+2/ 

1 

1 

1+^ 

^xyz', 


also 


10 
4 
3 

7 


17 
8 
8 
20 


13 
6 
1 
17 


=  124. 


Also  apply  the  proposition  to  show  that 


5  0  11 
8  7  2 
3  1  4 
8  1    0 


0 
20 

7 
6 


=  2188. 


EXAMPLES. 


45 


MISCELLANEOUS    EXAMPLES. 


1.   Find  the  value  of 


2  4      3      1       4      3 
-4      2-3      2  -1      2 

5_1      6      2-1      5 
1      11-2  -2-2 
7_3_5      1      4      2 

3  12-123 

1 

also  of 

12     22     14     17 
16-4       7       1  - 
10-3  -2       3  - 
7      12      8       9 
11      2       4  -8 
24      6       6       3 

20 
-2 
-2 
11 
1 
4 

10 
15 
8 
6 
9 
22 

2.    Expand  the  following: 

X      0      0    0  ...      0      a„ 
-1      a;      ic^  3^3...      a;"-ia„_i 

0-1      0    0  ...      0      a„_2 
0      0  -1    0  ...      0      a„_3 

1 

ai       62      0       0  ... 

^2     — ^1         ^3         0   ... 
ttg            0     — &2          64... 

rx4       0       0  -63... 

0 
0 
0 
0 

0 
0 
0 
0 

0      0      0    0  ...      0     a, 
0      0      0    0  ...  -1      ao 

a„       0       0      0...- 
a„+i    0       0      0  ... 

0 

1  K+i 

3.    Show  that 

1  0  0  0   aa;  +  /^2/  +  9^2 
0  10  0   7iaj4-  62/  +/2= 
0  0  1  0  gra;  -f  /y  4-  C2; 
0  0  0  1    ?a;4-m2/  +  w2; 
a;  2/  2;  1             h 

1 
0 
0 

X 

0 
1 
0 

y 

0 
0 

1 
z 

ax  -^  hy -\- gz -{- 1 
hx  -\-by  -\-fz  +  m 
gx  +  fy  +  cz  +  n 
k 

4.  Write  the  complementaries  of  the  following  minors  of 

|ao  61  02^3  64/5]:   10264!;   Ico/sl;  I  Co  6^4  62!;  1 61  63^4 1;   [d^esfi] 

5.  What  are  the  complementaries  of 

|ai2  (Xssl   and  |  ai2  ass  asg],    in  |  aoi  ai2  ass  «34  «45  ^sel? 


6.    Show  that 

0  1            1 

1  0        1+a 
1     a+1        0 

1     &H-1     6+a 
1     c+1     c+a 

1 

1+6 

a  +  6 

0 

c  +  6 

1 
1+c 
a-\-c 
6+c 

0 

=  23 

1+a     1        1 
1      1+6     1 
1         1      1+c 

46 


THEORY  OF   DETERMINANTS. 


7.   Prove  that 


\<h  h\  K  bs\  \(h  &4|. 

!«!  C2I  \ai  C3I  |ai  C4I. 

!«!  d2\  [ai  dal  |ai  d^]. 

\aik\  \cii  h\  \cii  k\' 


8.    Employing  the  notation 

/n\  ^  n{n-l)(n-2)  ...  (n-r  +  1) 
\rj~  rl  .        '■ 

show  that 


tti  a2  as  a^  ...  a^ 

61  &2  h  h  '"  K 
Ci   Cg    C3   C4   ...    c„ 
di  ^2  ^3  c?4  ...  dn 

1 

^n-2 
«1 

h  h  k  k  ...  L^ 

6.1 


a,  Z« 


rn 


^  rx-\-2y\        /x+2tj+l\   /x+2y+2\  ^^^  /x+3y-l\ 
Observe  that  (   ^   )-[      ^      j=(   ^_,    } 


/x-{-y+l\    /a;+2/+2\     /aJ+2/+3\ 

/aj+2/+2\    /x+y-^3\     ^x+y-{-A\     ^^^   /x+2y+l\ 


The  Product  of  Two  Determinants. 

V         50.    If  we  note  a  determinant  by  K,  and  another  by  i,  their 
product  P  is  evidently  expressed  by 

0  L 

The  form  of  this  product  suggests  the  probability  that  the 

product  of  two  determinants  may  be  expressed  by  writing  the 

factors  as  complementary  minors  of  a  determinant  of  higher 

order,  and  filling  in  the  vacant  places  due  to  one  or  both  of 


GENERAL  PROPERTIES  OF  DETERMINANTS. 


47 


the  factors  with  zeros.     Suppose,  for  example,  that  K  is  of  the 
third  order,  and  L  of  the  second  ;  then  P  would  take  the  form : 


«!   bx  Ci 

«£  h  C2 


a4  ft 
as   ft 


We  now  wish  to  discover  if,  when  we  fill  in  the  vacant  places 
due  to  K  or  L  with  zeros,  and  thus  make  P  a  determinant  of 
the  fifth  order,  P  will  still  be  the  product  of  K  and  L.  That 
this  is  the  fact  will  be  shown  in  the  next  article. 


51.  Theorem.  —  The  product  of  two  determinants^  K  and  i, 
of  degree  m  and  n,  respectively^  is  a  determinant  P,  of  degree 
m-{-n,  in  which  K  and  L  are  complementary  minors^  so  situated 
that  the  principal  diagonal  of  P  is  made  up  of  the  elements  in 
order  of  the  principal  diagonals  of  K  and  L  ;  the  vacant  places 
in  P,  due  to  either  K  or  i,  are  filled  with  zeros,  and  any  mn 
finite  elements  occupy  the  remaining  places. 


We  have  to  show,  for  example,  that 


KxL 


ai  &i  Ci 
CI2  ^2  ^2 
%  ^3  C3 

X 

a4  ^4  74 
as  ft  75 
a*  ft  76 

— 

ai  &i  Ci 

(I2    ^2    <^2 
«3    ^3    C3 

0  0  0 
0  0  0 
0    0    0 

a^  64  C4 

«5    ^5    C5 

ag  b^  c. 

a4  ft  74 
as  ft,  7s 
a«  ft  76 

=  p. 


(1) 


Developing  P  in  terms  of  the  elements  of  the  fourth  column 
and  their  complementary  minors,  we  have 


P  = 


0-4 


«!  bi  Ci  0   0 

—  as 

ag  62  C2  0   0 

ttg    63    C3    0     0 

tts  h  C5  ft  75 

«6    ^6    Co    ft  7c 

«!  61  Ci  0   0 

a2  62  C2  0    0 

as  63  C3  0   0 

^4  64  C4  ft  74 

<^6  h  ^6  ft  76 


+  a6 


«! 

^'l 

Ci 

0 

0 

^2 

&2 

C2 

0 

0 

«3 

&3 

C3 

0 

0 

^4 

h 

C4 

ft 

74 

«5 

h 

Co 

ft 

75 

or 


P  =  a4  Aa^  -  a^  A^s  +  og  Aae . 


(2) 


48 


THEORY   OF  DETERMINANTS. 


But      A, 


ft 


tti  bi  Ci  0 

ttg  62  C2  0 

ttg  63  C3  0 

«6  ^6  Ce  76 


-A 


«1 

h 

Cl 

0 

a^ 

h 

C2 

0 

as 

bs 

C3 

0 

as 

h 

Co 

75 

=  ftyel  «i  &2  C3 1  -  ft  75 1  tti  62  C3  I  =  ^  I  ft  75 

I  ft  76 
In  the  same  manner,  we  may  show  that 


K 


A  74 
ft  76 


also  Aa^  =  K 


ft  74 
ft  75 


Substituting  these  values  in  (2) ,  we  have 

since  the  second  factor  is  obviously  L,  expanded  in  terms  of 
the  elements  of  the  first  column. 

The  method  of  proof  here  given  is  perfectly  general,  and  is 
applicable  to  determinants  of  an}^  order.  Thus,  if  in  (1)  we 
make  y^  =  y^z=zO,  and  76=1,  P  takes  the  form  considered 
in  50.     The  student  can  readily  make  the  application. 

As  another  exercise,  the  student  may  show  that 


=  -\ciif2C3d^\x\€,beg7 


A=    ai  0    Ci  di  0  fi  0 
ttg  0    C2  (^2  0   /a   0 

as     0       C3     dg     0      /g      0 

^4    0     C4    ^4    0     /4     0 

0  65  0  0  65  0  9^5 
0  b,  0  0  e,  0  g, 
0    bj  0    0    ej  0  Qj 

"What  difference  would  it  make  in  the  result  if  tlie  zeros  in 
the  fifth,  sixth,  and  seventh  rows  of  A  were  replaced  by  any 
finite  elements? 

52.  Writing  the  product  of  |  aj  63  Cg  |  and  |  iCi  2/2  2^3 1 ,  in  accord- 
ance with  51, 

ai  61  ci  -  1       0      0     =  P, 
ttg  b.2  C2     0—1       0 

ttg     6g     Cg  0  0—1 

0    0   0      x^     yi     Zi 


0    0    0 
0    0    0 


OOo 


GENERAL  PKOPEKTIES   OF   DETEllMINANTS.  49 


we  have,  by  37, 
P  = 


0  0 

0  0 

0  0 

aiX^  +  a2yi-\-asZi  &!«!+ &2  2/i  +  &3  2'i 

a^x^+a^Vi  +  a^z^  biX2 -^  b2y2 -{- b^Zz 

ttl^S  +  «22/3  +  «32!3  b^Xs  +  622/3  +  b^Z^ 


which,  by  43, 

ai4  +  «2yi  +  «3^i     ^ 

«l»2  +  «22/2  +  "'    ^         '^ 

ai»3  +  42/3  + 


0  -10       0 

0  0-10 

0  0       0-1 

Cia^i  +  Cg  2/1  +  ^3%  xi     2/1     % 

^1^2    I    <^22/2  "r  <^3^2  **^2         2/2         % 

CiiBs  +  c^^/s  +  Cs^Js  a;''     2/3     ^3 


This  result  expresses  the  product  of  two  determinants  of  the 
third  order  as  a  determinant  of  the  same  order.  We  are  thus 
led  to  infer  that  the  product  of  two  determinants  of  any  order 
may  be  expressed  at  once  as  a  determinant  of  the  same  order. 

We  now  proceed  to  establish  this  important  multiplication 
theorem. , 

53.  Theorem.  —  The  product  of  two  determinants,  A,  A' 
of  the  nth  order  is  a  determinant  A"  of  the  same  order.  Any 
element  a^s  of  £i"  is  obtained  by  midtiplying  each  element  of  the 
rth  row  of  A  by  the  corresponding  element  of  the  sth  row  of  A', 
and  adding  the  products.* 

Before  giving  the  general  demonstration,  it  will  be  useful  to 
establish  the  proposition  for  the  product  of  two  determinants 
of  the  third  order,  and  note  carefully  the  form  of  the  result. 

*  Forming  the  product  by  columns,  the  statement  is,  of  course :  The 
element  in  the  rth  column  and  sth  row  of  A"  is  obtained  by  multiplying 
each  element  in  the  rth  column  of  A  by  the  corresponding  element  in  the 
sth  column  of  A',  apd  adding  the  products. 


50 


THEOKY  OF  DETERMINANTS. 


Put  A  = 


«1 

h 

Cl 

as 

62 

C2 

as 

63 

C3 

and  A'  = 


o-i  Pi  71 
«»  A  72 

^3    Ps    73 


Apxjl3*ing  the  theorem,  we  have  to  show  that 


AA'=A"  = 


«iai  +  &iA  +  Ci7i    «ia2  +  &i/?2+Ci72    aia3  +  6i/83+Ci73 

<^2ai  +  &2^1  +  C2  7l     «2a2  +  &2/52  +  C272     «2a3  +  ^2i83  4-C273 

%ai+63/?i+C37i  a3a2+63/J2+C372  a3a3-i-^3A+C373 


Since  each  element  of  A"  is  a  trinomial,  the  determinant  may 
be  decomposed  into  twent3'-seven  determinants  (35),  the  ele- 
ments of  which  will  be  monomials.  But  of  these  twenty-seven 
determinants  only  six  do  not  vanish.*  Those  determinants 
which  do  not  vanish  are  formed  by  taking  for  the  first  column 
a  set  of  first  terms  from  the  first  column  of  A",  for  the  second 
column  a  set  of  second  terms  from  the  second  column  of  A", 
for  the  third  column  a  set  of  third  terms  from  the  third  column  ; 
or,  by  taking  a  set  of  second  terms  from  the  first  column  of  A", 
a  set  of  first  terms  from  the  second  column  of  A",  and  a  set  of 
third  terms  from  the  third  column  of  A"  ;  and  so  on.  That  is 
to  say,  exactly  as  many  non-vanishing  determinants  can  be 
formed  from  A"  as  there  are  permutations  of  the  numbers  1,  2, 
3,  i.e.,  6.     Hence 


A''  = 

ttiai    61^2    C173 
«2ai    ^2(^2    C273 
«3<^i    hPi   C373 

+ 

«!«!     Ci72     5ift 

a-iai    C272    hfSs 
a^a^    Csy2    h/Ss 

+ 

61^1     ttittg    Ci73 
hPi    «2a2    C273 

bsPl      ttgOa      C373 

+ 

5l/?i      C172      dittg 

&2^1    C272    «2a3 
hPl    C372    «3«3 

+ 

Ci7i    ^A    CliOs 
C271    69  A    «2a3 

C37I     ^3^2     «3a3 

+ 

Ci7i    aia2    &ift 
C271  a2a2  b.2(Ss 

C371     ^302     Z^sft 

=  aiftys  («i 2^263)  -  ai/?372  (aib,c.i)  -  a^f^iys  {a.hc^) 

+  a3/5l72  («1&2C3)  —  a3/327l  («1^2C3)  +  a2ft7l  («1&2<^.0 

(by  30  and  31) 

*  It  is  obvious  that  the  determinants  formed  from  sets  of  first  terms 
taken  from  the  three  columns  of  A",  or  those  containing  sets  of  first  terms 
from  two  columns,  etc.,  must  vanish.  Similarly  for  determinants  formied 
from  sets  of  second  terms,  and  so  on. 


GENERAL  PEOPERTIES  OF  DETERMINANTS. 


51 


=  (cLihCs)  [ai/32 73+^2 /?3 71+^3 Ara—  a3/?2  7i—  a2A73  — aiftyg] 

=  («i^2C3)(aift73), 

which  establishes  the  proposition  for  the  special  case   under 
consideration. 


In  general,  let 


A  = 


«! 

h 

Ci 

^1 

^2 

b. 

C2 

4 

a. 

bs 

Cs 

^3 

ttn 

K 

Cn 

In 

and  A' 


«!  ft  7i  •••  ■^i 

«2    ft    72    •••     ^^2 

"3    ft    73    •••    ■^S 


ttn  ft    7»    •••    -^^ 


Then  the  product  AA'  =  A" 

«!«!  +  ^ft  +  Ci7i  +  ...  +  hXl 


«2ai-h&2ft  +  C2  7i  + 

<^3«l  +  ^3ft  +  C37i  + 


..+?2Ai 
..+^3^1 


Otn«l+  ^nft  +  C„7i+  ...  +  InK 
«ia2+^ft  +  Cl72  +  ...  +^1^.2 


a2a2  +  62ft  +  C272  4- 

a3a2  +  &3ft  +  C372  + 


..  -j-  /2A2     • 
..  +  ^3^2     . 


«ian+&ift4-Ciy«  + 

«2an4-&2ft+C2  7„  + 
«3an+&3ft+C37n  + 


a„a2  +  6„ft  +  c„72  4-.-.+^„  2   ...    «»an+^„ft+C„y„+...  +  Z„\ 

•  Now,  A"  may  be  decomposed  into  a  sum  of  7i~  determinants, 
the  elements  of  which  are  monomials.  But  it  is  obvious  that 
all  those  determinants  whose  columns  are  formed  from  sets  of 
first  terms  of  the  columns  of  A",  or  from  sets  of  second  terms, 
etc.,  will  vanish,  as  each  will  contain  identical  columns.  In 
fact,  all  those  determinants  into  which  A"  is  decomposed  will 
vanish  that  have  not  the  first  column  formed  from  a  set  of  kth. 
terms  from  the  first  column  of  A'',  the  second  column  formed 
from  a  set  of  rth  terms  from  the  second  column  of  A",  the 
third  column  formed  from  a  set  of  ^th  terms  from  the  third 
column  of  A",  and  so  on.  Now,  as  many  such  non-vanishing 
determinants  can  be  formed  as  there  are  permutations  of  the 
numbers  1,  2,  3  ...  n  ;  that  is,  n  !  Hence,  A"  is  decomposable 
into  n  !  determinants,  of  which  the  following  Ar  is  the  type  : 


62 


THEORY   OF   DETERMINANTS. 


bipi  kX2  aittg  . 

..  Ciy„ 

&2^1    ^2-^2    Ci20.3    • 

..  C27« 

^sA    ^3^    (^3<h    . 

•.  c^yn 

Ar  = 


PnPl    ^n^2    «»«3    •••    C„y, 

But  Ar  =  )8iA2a3  ...  y„  I  6i?2a3  •••  c„|. 

Now,  the  determinant  factor  of  A^  is  evidently  A  multiplied 
by  the  sign-factor  (  —  1)^,  in  which  p  is  the  number  of  inter- 
changes of  two  columns  which  must  be  made  in  A  to  leave  its 
columns  in  the  order  which  they  have  in  Ar.  Accordingly, 
Ar=(  — 1)^/81X203. ..y„A.  But  (  —  iy/3iX2<^3"'yn  is  a  term  of 
A',  since  the  number  of  interchanges  of  two  letters  which  must 
be  made  in  01^273  •••  K  to  obtain  the  arrangement  here  given 
is  p.  Accordingly,  Ar  equals  a  term  of  A'  multiplied  by  A. 
Thus  each  of  the  n\  determinants  into  which  A"  has  been 
decomposed  is  the  product  of  A,  and  a  term  of  A'.     .*.  A"=  AA'. 


Illustrations : 


12  0  3 

X 

0  110 

= 

1110 

3  2  11 

3  0  2  1 

10  0  1 

0  0  12 

2  110 

0+2+0+0  3+4+0+3 

0+1+1+0  3+2+1+0 

0+0+2+0  9+0+2+1 

0+0+1+0  0+0+1+2 


1+0+0+3  2+2+0+0 

l_|_0+0+0  2  +  1  +  1+0 

34-0+0  +  1  6+0+2+0 

0+0+0+2  0+0+1+0 


2  10  4  4 

=4 

15  2  2 

2    6    14 

2  6  14 

2  12  4  8 

16  2  4 

13    2  1 

13  2  1 

10  0  0 

X 

0  10  0 

= 

0  1  «  a 

1  0  a  a 

0  16^ 

10  6/? 

0  1  c  y 

1   0  C    y 

0  1                1  1 

1  a^  +  a^  ab  -{-  a/3  ac  +  ay 
1  ab  +  afi  b^  +  (i^  bc-\-l3y 
1  ac  +  ay  &C  +  )8y  (T  +  y" 


54.  Since,  before  multiplying  two  determinants  together,  we 
may  change  the  form  of  one  or  both  factors,  the  product  of  two 
determinants  can  be  expressed  in  a  variety  of  different  forms. 
As  an  illustration,  the  student  may  verify  the  following  equa- 
tions : 


GENERAL  PBOPERTIES  OF  DETERMINANTS. 


53 


kxA 


I  a2'^02 


aitti  +  aq/?! 
^itti  +  ^aA 

(Xi  ttj  -|-  012  012 

6iai  +  &2a2 


a2a2 +  12/82 

ttittg  +  ^2  A 
6ia2  +  &2/52 

«2  A  +  &2/S2 

«l/?l+«2A 
5lft  +  62/?2 


EXAMPLES. 


1.    Show  that  one  form  of  the  product  of 


1  aa" 

X 

a^-a  1 

is 

1  6  &2 

6^-6  1 

1  c  c^ 

c'-c  1 

W-ch  +  c"  (? 


2.    One  form  of  the  product  of 


a+6  c  c 
a  6+c  a 
6         6      c+a 


-i& 


(? 


a" 


&2 


IS 


3.   Find  the  product  of 


a  a  a  a 

and 

ahhh 

a  b  c  c 

abed 

-110  0 
0-1  1  0 
0     0-1      1 

111-1 


,  and 


thence  show  that  the  first  determinant  =  a  (b—a)  (c—b)(d—c), 
4.   Show  that 


1111 
-1-1  1  1 
-1  1-1  1 
-1      1      1-1 


a 

h 

c 

d 

X 

h 

—a 

d 

c 

c 

d 

—a 

b 

d 

c 

b 

—a 

b-\-c-i-d—a 
b—a-\-d-\-c 
c+d—a-\-b 
d-^c-{-b—a 


a—b-{-c+d 
■  b-^a-{-d-\-c 
■c—d—a+b 
d—c+b—a 


a-\-b—c+d 
—b—a—d-\-c 
—c-\-d-\-a+b 
—  d-\-c—b—a 


a+b-\-c—d 
—  b  —  a-\-d—c 
—c-}-d—a—b 
—d-\-c-\-b-\-a 


54 


THEORY  OF  DETERMINANTS. 


=  (6+c+cZ-a)  (c+d+a—b)  (d-j-a+b-c)  {a-\-b-\-c—d) 
1111 


1 

1 

-1 

-1 

1 

-1 

1 

-1 

1 

-1 

-1 

1 

and  thence  show  that  the  first  determinant 
=  (b-\-c-\-d—a)  (c-i-d+a-b)  (d+a+b—c)  (a+b+c—d), 
5.    Show  that 


1  a^+a^  -2a  -^2a 

1  b^-\-l3^  -26  -2(3 

1  c2+y2  _2c  -2y 

1  d2+82  -2d  -28 


a^+a^  1  a  a 
b^+fS'  1  b  /3 
c^+Z  1  c  y 
^2+  8M  d  a 


0 

{a-cy-{-{a-yr 
la-dy-\-(a-8y 


(a 


0 

(b-cy+(i3-yy 

0 

(c-dy+(y-sy 


(a-dy+(a-&y 
(b-dy+(^^By 
(c-dy+{y-By 

0 


6.    Show  that 

ai  5i  Ci  1 

X 

1  0  0  fci 

X 

10  0  0 

ttg    &2    ^2    1 

0  1  0  A;2 

0  10  0 

as  63  C3  1 

0  0  1  A:3 

0  0  10 

1110 

0  0  0    1 

hi  Jiz  /is  1 

equals  the  determinant  in  Example  12,  page  38. 
7.   Find  the  two  determinant  factors  of 


«i  Mi+Ci2/i  ^2+^12/2  ;  also  of 

«2  b<^i'-{-c^i  62^2+^22/2 

Cts  Ml  +  Cs^l  &3aJ2+C3y2 

8.  Form  the  product  of 


%  61 

Og    62 


axi-\-cZi  0  fxi+gzi 

ax2+by2-\-cz2  dy^  fx2+gz2 

bys+czs  dys        gz^ 

and 


ai  A  ri 

"2   A    72 
as   /?8   78 

The  order  of  the  first  determinant  may  be  raised  to  that  of 


the  second  by  writing  it 


0^2    ^2    ^2 

0    0    1 


and  the  product  can  then 


GENERAL   PROPERTIES    OF   DETERMINANTS. 


55 


be  found  in  the  usual  way.  If  we  wish  the  product  to  contain 
only  the  elements  found  in  the  two  factors,  how  should  the  first 
determinant  be  written? 

From  this  example  it  is  evident  that  the  product  of  any  num- 
ber of  determinants  of  different  degrees  can  he  expressed  as  a 
determinant  of  the  nth  degree.^  n  being  the  highest  degree  among 
the  factors. 

9.    Emplc 


\  —  c-\-id 
may  be  written 


oying  the  notation  i  =  V—  1,  shaV  \bsX  tiie  product  of 
a-\-ih     c-j-id    and  [     ai—ihi     Cj  — idJ 


a  —  ib 

I     D-iC 
-B-iA 


\-Cy 

B^iA 
D  +  iC 


C  =  abi  —  aib  +  cdi  —  Cid 
D  =  aa-i  +  bbi  +  cc^  -}-  ddi ; 


m  which 

A  =  bc-i  —  biC  +  adi  —  aid 

B  =  cai  —  c^a  +  bd^  —  bid 
and  thence  show  that  the  product  of  two  sums,  each  of  four 
squares,  is  itself  the  sum  of  four  squares,     (Euler's  Theorem.) 

10.  Show  (1)  that  the  product  of  {aiboCs]  and  \piq2Ts\  may 
be  expressed  as  a  determinant  of  the  fourth  order  by  writing 
the  two  factors    cij  bi  Cj  0    and  —  i^i  ^i  0  rj    respectively. 


cii  bi  Ci  0 

and  — 

a2  62  C2  0 
«3  h  C3  0 
0    0    0   1 

Pi  Qi 

0 

n 

P2  q2 

0 

^2 

Ps  Qs 

0 

^3 

0    0 

1 

0 

(2)  By  writing  the  two  factors 

Pi  0  0  qi  n 
P2  0  O  q2  rg 
Ps  0  0  qs  Vs 
0  10  0  0 
0  0  10  0 
show  that  the  product  is  a  determinant  of  the  fifth  order. 

(3)  By  writing  the  two  factors 


«!  bi  Ci  0  0 

and 

^2    62    C2    0    0 

as  bs  Cs  0  0 

0    0    0    10 

0    0    0   0  1 

tti  61  Ci  0  0  0 

and 

aa  62  C2  0  0  0 

asbsCsO  0  0 

0    0    0   10  0 

0    0    0   0  10 

0    0    0   0  0  1 

10  0  0    0    0 

0  10  0    0    0 

0  0  10    0    0 

0  0  0  piqiri 

0  0  0  P2  qz  r2 

0  0  0  Ps  qs  n 

*  SI 


66  THEORY   OF   DETERMINANTS. 

show  that  the  product  is  a  determinant  of  the  sixth  order. 
This  example,  and  the  theorems  of  51  and  53,  show  that  the 
product  of  tivo  determinants  of  the  nth  order  can  be  eoopressed  as 
a  determinant  of  each  of  the  following  orders:  nth,  (n  +  l)^A, 
(w+2)«7i...  (2w-l)«7i,  2nth, 

55.  Theorem.  —  Any  determinant  A  may  be  expanded  as  a 
sum  of  products  of  pairs  of  minors.  The  first  factor  of  each 
product  is  a  minor  of  the  rth  degree,  formed  from  a  set  of  r 
chosen  rows,  and  the  other  factor  is  the  complementary  minor  of 
the  first  factor .  The  sign  of  a  product  is  -f-  or  — ,  according  as 
the  product  of  the  ^jrincipal  terms  of  the  factors  regarded  as  a 
term  of  A  is  -j-  or  —  .* 

Every  term  of  A  contains  r  elements  from  the  columns  of  a 
set  of  r  columns  found  in  the  n  columns  and  first  r  rows  of  A. 
That  is  to  say,  from  every  minor  of  the  rth  degree  formed  from 
the  first  r  rows,  r  I  partial  terms  of  A  can  be  formed.  Now, 
the  remaining  (n—r)  elements  of  every  such  partial  term  will 
be  found  in  the  remaining  rows  and  columns  after  removing 
one  of  these  minors  of  the  rth  degree.  Or,  in  other  words, 
(n— ?')  !  partial  terms  of  A  corresponding  to  the  r !  other  par- 
tial terms  are  found  in  every  minor  complementary  to  one  of 

the  first  set.     — ; — minors  of  the  rth  degree  can  be  formed 

r  !  (n— r)  ! 

from  the  first  r  rows.  Now,  the  product  of  two  such  comple- 
mentary minors  gives  r !  (n— r)  I  terms  of  A  ;  consequently,  the 
sum  of  all  the  products  gives  n  !  terms,  i.e.,  the  full  number  of 
terms  in  A.- 

To  fix  the  sign  of  any  product  in  this  expansion,  we  have 
only  to  remember  that  its  sign  must  be  the  same  as  the  sign  of 
the  product  of  the  principal  terms  of  the  two  minors.  This 
latter  product  being  a  term  of  A,  the  sign  of  the  product  of  the 
two  minors  must  be  the  sign  of  the  product  of  their  principal 
terms,  regarded  as  a  term  of  A. 

If  the  selected  rows  are  not  the  first  r  rows,  we  can  easily 
make  them  so  ;  then,  after  giving  A  the  proper  sign  factor,  the 
demonstration  applies  as  given. 

*  Tills  expansion  is  knowr  *^  Tiaplace's  Theorem. 


GENERAL  PROPERTIES  OF  DETERMINANTS.     57 

Illustrations : 

Selecting  the  first  two  rows  in  ki  h^  c^d^]-,  we  have 

4- 1  6i  Cgl  ^3^4!  —  I &i  cZgl  I tto  C4I  4- 1  Ci  dgl  I CI3&4I . 
Let  the  student  select  the  first  two  columns  of  1%  62  C3  (^4! ,  and 
expand,  obtaining 

\(hh\\c3di\  —  \aM\c2di\-{-\aM\C2ds\-h\a2bs\\cM 
—  lazb^WcM-hldsbiWcidzl. 
Show  that 
loihcsd^esl  =  —  lazb^l  \cidse5\-{-\a.jC^\  Ibidse^l  —  laid^l  IbiC^e^l 
+  \a2ei\  16103^51  —  16204!  laid^e^l  +  lbzd^l  la^c^e^l 
—  16264!  !aiC3d;5!  + 10264!  Iai63d5l4-!c2e4l  laib^dgi 
-Id^e.l  \a,bsc,\.  >  !^^/^y/^SvyS' 

What  is  the  relation  of  41  to  the  present  theorem?   .  -'     * 

56.  It  will  be  interesting  to  note  what  results,  if,  instead  of 
multiplying  the  minors  of  the  rth  degree  formed  from  r  chosen 
lines  by  their  complementaries,  as  in  the  last  article,  we  mul- 
tiply every  such  minor  by  the  complementary  of  a  corresponding 
minor  formed  from  r  lines  different  from  those  first  chosen. 
By  the  preceding  article 
jai  6203^465! 

=  !ai62l  !c3d4e5!  —  !ai63l  \c2d^e^\  +  !ai64!  \c2dse5\  —  \aM  \c2d.^ei\ 

+  ka  63!  !cid4e5!  —  |a264!  101^365! +  |a265l  101^364! -I- |0364l|cirZ2e5l 

—  Ia365l  !cid!2e4!  +  !«46^  IcidzS^l. 

Now,  if  in  the  above  we  write  c  for  6,  it  is  evident  that  the 
determinant  on  the  left  vanishes,  and  hence  the  second  mem- 
ber vanishes  ;  but  by  this  substitution  we  multiply  the  minors 
formed  from  the  ^?'s^  and  third  columns  of  I  a^  62  03^/465 1  by  the 
complementaries  of  the  corresponding  minors  formed  from  the 
Jirst  and  second  columns.  It  is  obvious  that  the  truth  here 
exemplified  holds  in  general.  Moreover,  it  includes  the  special 
case  of  45. 


68 


THEORY   OF  DETERMINANTS. 


In  symbols,  the  expansion  of  a  determinant  by  55  is  expressed 


by  writing 


A  =  :S|ap6J  Aflp,6^, 


where  the  chosen  columns  are  two  in  number,  or 

A=:Slap&gCj  Aaj„6j,c„ 

where  the  chosen  columns  are  three  in  number ;  and  so  on. 

Employing  the  notation  of  double  subscripts,  we  have,  in 
general, 

57.  Theorem.  —  The  product  of  a  determinant  A  =  I  ai„| ,  and 
any  one  of  its  minors  M,  of  order  m,  is  a  determinant  A'  of 
order  n  -\-  m.  A'  is  expressible  as  the  sum  of  pi'oducts  of  pairs 
of  minors  of  ^;  the  first  factor  of  each  product  is  a  minor  of  A, 
formed  from  r  chosen  rows  containing  J/,  and  the  second  factor 
is  that  minor  of  A  containing  the  complementary  of  the  first 
factor  and  the  minor  M.  The  sign  of  each  product  is  determined 
as  in  55. 

Let  the  chosen   rows   referred   to   in  the  statement  of  the 
theorem  be  the  first  r ;  then,  by  51,  we  have  at  once 
A'  = 


«11      %2     .-.^l*-! 

«1A      .-.air-l 

Ctlr 

«lr+l       • 

••«!« 

0   .. 

.         0 

0 

ttai      %2     "-ailc-l 

«2Jt      ...agr-l 

a2r 

«2.+l       • 

..a,,       0   .. 

.     0 

0 





... 

«Ar+l      • 

..a,„       0   .. 

.     0 

0 

^Al       <^A2      '"^kk-\ 

a,ac      •••«*r-l 

a*r 

<^k+l\(^k+\J'"(^k-h\k-l 

(^k+yc  • '(^k+i/-- 

-1  ^ft+lr 

^A+Ir+1* 

..a.+i.    0   .. 

.     0 

0 





... 

... 



.    ... 

... 

^r-ll^r-12'"^r-lA:-l 

Ov_^...a,_i,. 

-ittr-lr 

«y-ir+i. 

..«.-!„      0     .. 

.     0 

0 

a,l      a^      "'Ctrk~l 

aru    ...a.r-1 

a„ 

a^r+i    ' 

.     0 

'Ctr+lr- 

0 

1  ^r+lr 

^r+uf^r+l2"  '(^r+lk~l 

Ctr+lk-"<^r+lr- 

-l«r+lr 

» 



... 

... 



.       ... 

... 

«nl      an2     —Clnk-l 

0    ...     0 

Ctnr 
0 

ttnr+l      • 
0           . 

-a^n 

Clnk      '. 

-a^r-i 

«nr 

0        0       ...        0 

..    0 

a^    .. 

'C(kr-1 

a*r 

0        0       ...        0 

0    ...     0 

0 

0           . 

..   0 

Ctk+lk" 

'dk+lr. 

•iCti+lr 





... 

... 





.       ... 

... 

0     0     ...     0 

0    ...     0 

0 

0       . 

..    0 

Clr-lk" 

•«r-lr- 

-iCt'r-lr 

0     0     ...     0 

0    ...     0 

0 

0       . 

..   0 

ark    •• 

•«rr-l 

«rr 

GENERAL  PROPERTIES  OF  DETERMINANTS. 


59 


where  the  minor  by  which  A  is  multiplied  is  enclosed.  Further, 
observe  that  the  n  —  r  rows  of  A  not  included  in  the  chosen 
rows  are  prolonged  in  A'  with  the  elements  of  these  same  rows 
repeated  in  order  of  the  columns  beginning  with  the  kth.  Now 
add  the  A;th  row  of  A'  to  the  {n  +  l)th,  the  (fc  -f-  l)th  row  to 
the  {n  +  2)th,  and  so  on,  finally  adding  the  rth  row  to  the  last. 
Afterward  subtract  the  (n  +  l)th  column  of  A'  from  the  A:th, 
the  (w  +  2)th  column  from  the  (A;+l)th,  and  so  on,  finally 
subtracting  the  last  column  from  the  rth.     Then 


«11 

012 

...ttli-l 

f'l* 

...ai^_i    Oi^ 

%r+l 

...«u 

«21 

022 

•••«2*-l 

«2* 

...a.2r-l       Oj2r 

«2r+l 

...asn 

%1       ^*2       "'^kk-l 
^r+1 1  ^r+1 2  •  •  •  ^r+1  A-1      ^ 

<^r-ll<^r-12"'<^r-l&-l       ^ 


«'U      .• 

.a^r-i 

«■*. 

a^+u- 

•«mr- 

-1  (^k+\r 



.         ... 

... 

«,_!,.. 

.a.^ir- 

-l«,-lr 

a,,     .. 

.a,,_, 

a,. 

f'ftr+l 


''A:+ln 


^r-lr+l'"^r-l»      ^ 


■r+1 


0 
0 
0 

0 
0 
0 

... 

... 



0 
0 

0 
0 

0       ... 

0 

0 

0       ... 

0 

0 

0    ... 

0 

0 

0    ... 

0 

0 



... 

... 

0    ... 

0 

0 

0    ... 

0 

0 

a.+ii...o. 

+lr- 

l«r+l,- 



... 

... 

«nt      •••«« 

r-V 

(^nr 

(^kk      •••«* 

r-1 

Ctkr 

<^t+lA*"<^A+lr- 

l«A+lr 



... 

... 

ar-W'Clr 

-Ir- 

l«r-lr 

a,j,    ...a. 

r-1 

a,. 

By  55  A'  can  be  decomposed  into  products  of  pairs  of  minors, 
viz.,  the  minors  of  the  rth  order  formed  from  the  first  r  rows 
and  their  complementaries.  Since  the  elements  in  the  columns 
of  A'  directly  below  M  are  zeros,  all  the  minors  of  the  rth 
order,. formed  from  the  first  r  rows,  will  have  complementaries 
that  vanish  unless  the  said  minors  contain  the  given  minor  M. 
Hence  the  first  factors  of  the  products  in  the  expansion  of  A' 
will  all  be  minors  of  A,  of  the  rth  order,  that  contain  the  given 


60 


THEORY  OF  DETERMINANTS. 


«1 

&i 

Ci  di 

6,0 

0 

^2 

&o 

C2  (^2 

^2  0 

0 

a-i 

bs 

Cg  <^3 

63  0 

0 

a. 

h. 

C4  d^ 

64  64 

C4 

as 

h 

C5  d. 

65  h 

C5 

0 

0 

0  0 

0  62 

C2 

0 

0 

0  0 

0  63 

C3 

minor.  Further,  each  complementary  of  such  a  factor  is  made 
up  of  the  n  —  r  rows  of  A  not  found  in  the  first  factor  and  the 
r  —  Tc  +  l  rows  in  which  M  is  found.  Which  proves  the 
theorem. 

I«1&2C3C24  65!  1^203!  = 

tta  b^  C3  0^3  63  0    U 

a^  64  C4  (^4  64  64  C4 

1^162^31  \d^e^h2C^\ 
16102^3!  I a4 656203! 
16102631104^562031  ; 

!«!  62  63(^4  65!  16203(^4!  =  10162630541  1656203^4!  +161020^3641  10562  03^41. 

The  student  may  show  (change  the  rows  into  columns  before 
applying  the  theorem) 

1016263^465!  I63C4I  =  1016263^4!  1636465!  — 1016364(^5!  I636462I 

+  1026304(^5116364611; 

1016263^4651(^2 
=  161(^263!  la4&6C?2l  —  \cid2e^\  \a^hsd^\  +  161(^265!  10364(^21 
+  l62C?364l  loi65c22l  "  \<^2di^5\  lo, 64(^21  +  Uid^e^l  \aibMi 

=  —  1^162!  1036465(^2!  +  1(^263!  !«! 6465(^2!  —  1(^264!  IOi6365C?2l 

+  lc?2e5ll«i  6364(^2! » 


The  second  illustration  given  is  especially  interesting  as  it 
shows  the  form  of  the  product  when  the  minor  is  of  order  n  —  2. 
In  that  case  the  chosen  rows  are  7i  —  1  in  number,  and  the 
development  consists  only  of  two  terms,  each  term  being  the 
product  of  two  determinants  of  the  (71— l)th  order.  If  we 
change  the  order  of  rows  and  columns  in  the  result,  we  have 

!«!  6263(^4  65!  16263(^4!  =  10162  63(^4!  16263(^465!  — 16162(^36411026364(^5!, 

or  A   Aai,e6  =  Ae5  Aa^  —  Aa^  Ag,  ; 

and,  in  general, 

A   Aa,7fc,  apq  =  Aart  ^Opq  —  Aa,-j  Aop*. 

Employing  an  obvious  extension  of  the  notation  described  in 
the  latter  part  of  39,  the  last  formula  becomes 

A      ^^^      =  i?^  ^  _  ^  -^ 
dOi 


dcii^  da        da^  da 


-iq   ^%k 


GENERAL  PROPERTIES  OF  DETERMINANTS.     61 


Rectangular  Arrays  or  Matrices. 

68.  As  a  determinant  is  a  function  of  n^  quantities,  the 
elements  are  always  found  in  a  square  array.  It  is  often 
necessary  to  consider  the  determinant  obtained  by  applying  the 
process  of  53  to  two  rectangular  arrays  of  elements,  i.e.,  arrays 
in  which  the  number  of  rows  is  not  equal  to  the  number  of 
columns.     We  will  now  investigate  the  value  of  this  product. 

1st.  When  the  number  of  columns  exceeds  the  number  of 
rows: 

The  product  of  two  arrays  (matrices)  of  elements  in  whicli  the 
number  of  columns  (m)  exceeds  the  number  of  rows  (n),  is  a 
determinant  which  is  equal  to  the  sum  of  all  the  products  in  which 
the  first  factor  is  a  determinant  of  the  nth  order  formed  from  the 
first  array  (matrix) ,  and  the  second  factor  is  the  corresponding 
determinant  of  the  nth  order  formed  from  the  other  array 
(matrix) .     Let  the  two  arrays  of  elements  be 

«ii  «i2  •••  <^in  •••  ^im  1       an  ai2  •••  ^in  •••  aim  1 

Cl2i    a^   ...   a2n   . . .    a2m     I        g^jj  J      a21    0-22   •  •  •   a.2n   •  •  •   CL2m     I  n  <1  m 

<^nl    «'m2  •••   Ct„„  ...    «„„,  J  a„i    a„2  •..   Clnn  •••   CLnm    J 

Applying  the  process  of  53,  we  have  the  determinant 

^nau4-*'*+<^'l»ainH h^lw  "im  <^llf^21-\-"--\-CLln<hn 

<^21  an  H h  a2«ai„  H f-  «2m  ai^  ^21  a2i  H h  Of 2na2n 

<^niaiiH httn^a^.H ha„„ai„        ania2iH ha^^og^ 

H \-(^imf^2m   •'•   «na„iH |-ai„a„„H \-ai^a„„ 

-h"'-^-Cl2ni"-2m    "'    «2ia„iH \- Ci2n  O-nn -\ 1-«2„  a„„ 

H \-(^nm<^2m    '"     «„ia„iH \-Ctnn"-nn-\ \- <^'nm  O-n,, 

Now  we  may  form  from  A  a  number  of  determinants  A,,  Ag,  A3  ••• 
of  the  nth  order,  the  elements  of  which  are  all  polynomials  con- 


62  THEORY   OF  DETER^UKANTS. 

sistiug  of  n  terms  each.     The  number  of  such  determinants  is, 

of  course,  m(m  -  1)  (m-2)  -  (m-«  + 1).     ^et  us  consider 

n! 
one  of  these  determinants  ;  take,  for  example  Ai,  whose  columns 
are  formed  from  the  first  n  terms  in  the  columns  of  A.     AVe 
have,  accordingly, 

Ai=  auan4-C^i2ai2-| h^inain 

«2ian  +  <^22ai2H |-«2nain 

^n\  "11  +  0tn2  ^12  H f"  ^nvP-\n 

<^lia21  +  <^12«22H \-(^\nf^2n    '"     Ctji  «,»!  +  « 12  an2H h^lnOnn 

«21  «21  +  ^22«22+*"+^2na2»     •*"     ^<21  a„i  +  a22  a„2 H |-«2»ann 

«nia21  +  <^n2a22H \-<^nnO-27i    '"     «»ianl  +  ««2aH2H h^nnttnn 

Now  Ai  is,  by  53,  the  product  of  two  factors,  the  first  of  which 
is  the  determinant  formed  from  the  first  n  columns  of  the  first 
array  of  elements,  and  the  second  is  the  determinant  formed 
from  the  corresponding  n  columns  of  the  second  array.  In  a 
similar  manner  we  may  show, that  each  of  the  determinants 
Ai,  Aa,  Ag---  is  the  product  of  two  factors,  each  factor  being  a 
determinant  formed  from  n  corresponding  columns  of  the  two 
given  arrays.     Then  in  order  to  establish  the  proposition  it 

remains  to  be  shown  that   A  =  Ai  +  Ag  +  Ag  H .     Each  of  the 

determinants  Ai,  Ag,  Ag---  can  be  decomposed  into  n!  non- 
vanishing  determinants  whose  elements  are  monomials.  Ac- 
cordingly the  sum  Aj  -j-  As  +  AgH —  will  contain 
m(m— l)(m  — 2)  ••-  {m  —  n-{- 1) 
non-vanishing  determinants  whose  elements  are  monomials. 
Returning  to  A,  we  see  that  it  can  obviously  be  decomposed 
into  m"  monomial  element  determinants ;  but  those  which  do 
not  vanish  are  only  m(m  —  l)  (m— 2)--- (m  —  n-f  1)  in  number. 
Now  observing  that  each  one  of  these  monomial  element  deter- 
minants is  a  jmrt  of  that  one  in  the  series  Aj,  A2,  Ag---  in  which 
its  columns  occur  as  parts  of  columns,  the  proposition  is  estab- 
lished. 


GENERAL  PROPERTIES  OF  DETERMINANTS.      63 

Illustration : 

Performing  the  operation  of  53  upon 

ag  02  Co  0,2  j  a2  P2  72  f>2  j 

we  obtain  the  determinant 

«2ai+  ^2A  +  C2yiH-(^2Sl        «2a2  4-&2ft +^^272  +  ^2^2 

This  determinant  the  student  can  readil}^  show  is  equal  to 

(«i  W  («1  A)  +  (^^2)  (aiy2)  +  («id2)  (ai82) 
+  (2^1  C2)  (A  72)  +  (&lC^2)  (/?lS2)  +  {cA)  (7182)  . 
2d.    Wlie7i  the  number  of  rows  exceeds  the  number  of  columns. 
Consider  the  two  arrays. 

a^  &2  f    and  ttg  ^2 

Multiplying  as  before,  we  have 

ajai  +  by^^     aia2  +  &1/32     ajag  +  ftjft  =  0. 

«2ai  +  i^2/?l        «2a2  +  &2/?2       <^2  ^3  +  h(^S 
Cts"!  +  ^sA       «3a2  +  ^sft       WsCtg  4-  63^3 

The  value  of  A  is  readily  seen  to  be  zero  when  we  notice  that 
it  can  be  obtained  by  multiplying  two  determinants  formed  from 
the  two  given  arrays  by  prefixing  a  column  of  zeros  to  each.  The 
method  of  proof  employed  in  this  special  case  is  general.  It  is 
only  necessary  to  add  to  each  array  as  many  columns  of  zeros  as 
are  necessary  to  make  each  array  square,  and  then  compare  the 
product  of  the  two  determinants  thus  formed  with  the  deter- 
minant formed  by  compounding  the  two  matrices. 

Reciprocal  Determinants.* 

59.  If  the  principal  minors  of  the  elements  of  a  determinant 
are  themselves  made   the  corresponding  elements  of  another 

*  Reciprocal  determinants  would  more  properly  be  considered  in  the 
next  chapter  since  they  are  among  the  "  special  forms,"  but  for  several 
reasons  it  is  thought  best  to  introduce  them  here. 


64 


THEORY   OF  DETERMINANTS. 


determinant,  the  determinant  thus  formed  is  called  the  reciprocal 
or  adjugate  determinant.  Or,  in  other  words,  the  elements  of 
the  reciprocal  determinant  are  the  complementary  minors  of  the 
corresponding  elements  in  the  original  determinant. 


The  reciprocal  of  {cti  h^  c^)  is 


ipp 


(as  Cg)  (tto  h) 

KC3)      -(ai^a) 
(ai  C2)  (ai  62) 


Assimilating  the  notation  of  19,  we  have 

\AiB2Cs,..L,\,  1^1,1,  or  1^11  ^22 -433... A„l, 

for  the  determinant  adjugate  to 

I  «i  h  C3  ...  ^n  I?  I  «i^  |,  or  1  an  a22  %3  •.•  <^nu|? 
respectively. 

If  the  minus  signs  in  the  first  illustration  are  erased,  what  is 
the  effect  upon  the  determinant?     How  is  it  in  general? 

60.    Theorem.  —  The  determinant  A'  adjugate  to  any  deter- 
minant A  oftJie  nth  degree,  equals  the  (n  —  l)th power  of  A. 

We  have,  for  example. 


Whence 


A  = 

«!    61    Ci 

^2    ^2    C2 

«3    h    C3 

AA'  = 

A  0  0  = 

0  A  0 

0  0  A 

and  A'= 


A^ 


A  B,  C\ 

A2  B,  (72 

Ao     Bo      C.Q 


The  process  here  exemplified  is  perfectly  general,  hence  the 
proposition. 

61.  Theorem. — Any  minor  of  the  kth  degree  of  the  reciprocal 
determinant  A'  is  equal  to  the  complementary  of  the  corresponding 
minor  in  the  original  determinant  A  multiplied  by  the  {k  —  \)th 
power  of  A. 

Let  A  =  I  ai4 1,  and  A'=  I  ^14 1 . 


GENERAL  PBOPERTIES  OF  DETERMINANTS. 


65 


Transform  A  and  A'  so  that  the  minors  |  ctn  ^32  «44  ]  and 
I  All  -^32  ^44  I  occupy  the  first  three  rows  and  columns  in  their 
respective  determinants.     Then 

A=(  — 1)''  ail     «i2 

^31        ^^32        ^ai 
41  42 


and  A'=(-l)'^ 

An 

Ai, 

^4 

^1, 

A. 

As, 

^4  34 

^33 

-I41 

^442 

-i« 

^« 

An 

A.^_ 

A.^ 

^.. 

Then 

■ 

lAl    A2    ^44|=(-l)'^ 

Ai 

Ay2 

A,, 

A„ 

Ai 

A,, 

Au 

•^33 

A,i 

Aj^ 

Au 

^43 

0 

0 

0 

1 

Multiplying, 

A\Aii  A^  A^\=z 

A 

0      0 

«13 

=   a2C 

0 

A     0 

«83 

0 

0      A 

"c 

0 

0      0 

a^ 

Whence  |  An  ^433^44  ]  =  aog  A^,  which  is  the  required  value  of  a 
first  minor  of  A'. 

To  find  the  value  of  a  second  minor  of  A'  we  may  proceed  as 
follows : 


The  minor 

1-^22  As, 


(-l)nA2 

•^32 

0 
0 


-"■2; 

A 
0 
0 


■^21 

A^i 
1 
0 


A4 

-^34 

0 

1 


and  the  corresponding  form  of  A  is 

(-1) 


«'23 

a^i 

a24 

«33 

asi 

(X34 

^'13 

«n 

«14 

«43 

«41 

«44 

QQ  THEOBY  OF   DETEEMINANTS. 

As  before, 


A  Moo  AJ  = 


A^  ail  a. 


A    0     0821    «2-  ,     _ 

0  A  agi  a34 
0  0  an  an 
0    0   a^i  a^ 

Whence,  |^22  ^33!=  \<^n  «44|^- 

The  student  may  put 

A  =  I  ai  62  C3  di  I  and  A'  =\Ai  B^  C^  A  | » 

and  then  show  that 

\B,  a,  A|=aiA2; 

Ml  A|=  I&2  C3IA. 

The  general  theorem,  of  which  the  preceding  are  special  cases, 
is  proved  as  follows  : 

Let  A  =|ai^|  and  A'=  |An|? 

and  let  the  minor  of  the  Mh  order  of  A'  whose  value  is  sousfht  be 


V= 


^PiQi   ^PiQz   ^PiQs    •' 


^PiQk 


^PiQl     ^P2l2     ^P2Q3      '"      ^P2<lk 

\^PkQi  ^Pk<l2  ^PkQa  '"  ^PkQk^ 

Now  putting 

fi  =Pi  +P2  +  •••  +Pu  +  qi  +  q2-\ f-  ^*, 

we  may  write 

Qp^q^  ...  Op^g^  ap^i  ...  Ctjy^q^-l  Ctpiq^+l  •••  <Vi?2-l  ^Pi?2+1  " 
^PiQi  '"  ^PiQk  ^Pi'i-  '"  ^P2Qi-l  ^zQ-i+l  •••  ^i>2(?2-l  ^i'2Q'2+l  •' 

^PkQi  •••  ^PkQk  ^Pk^'"'  ^i'ftffi-l  ^PkQi+'^  '"  ^Pk^2-^  ^*ff2+l  •' 


Cip^n 


^Pk^ 


aiq^  ...  aig-^   an  ...  a\q^-\  aig-^+i   ...  aig^-i   ai^^+i   ...  ai» 

a2g^    ...   a2g^     ttsi    ...  a2g'j-l    O^g^i+l     •••   ^2^2-1     ^ffa+l     •••   «2n 


*  In  this  determinant  the  subscripts  }\,  }\,  p^,  ...  g^,  q^,  q^,  ...  of  course 
stand  for  any  integers  in  order  of  magnitude. 


GENEEAL   PROPERTIES    OF   DETERMINANTS.  67 

The  coiTesponding  form  of  A;^  is 

. . .  Ap^q^i  Ap^q^+i . . .  Ap^n 

•  •  •  ■4^2?2-l  Ap^q^+l  .  .  .  ^^271 

•  •  •  Ap^q^-1  Ap^q^+i  . . .  Ap^n 
...  0  0         ...       0 

...       0  0      ...    0 


ApiQi  ' 

•  •  Ap^qi^ 

Axl" 

•  Ap^q^- 

-lAp,q,+l 

Ap^Qi  ' 

••Ap^q„ 

Ap^\'' 

'  Ap^q^~ 

-1  Ap^q^+i 

Apj,gj_  • 

'  •  Ap^q^ 

Aa-1" 

Apf^Qi- 

-i4pftgi+i 

0     . 

..     0 

1      .. 

0 

0 

0     . 

.     0 

0    .. 

0 

0 

0     . 

.     0 

0    .. 

1 

0 

0     . 

.     0 

0    .. 

0 

1 

0     . 

.     0 

0    .. 

0 

0 

0     . 

.     0 

0    .. 

0 

0 

0     . 

.     0 

0    .. 

0 

0 

0 

0      . 

..    0 

0 

0      . 

..    0 

1 

0      . 

..    0 

0 

1    . 

..    0 

0 


•••l(n-fc) 


We  notice  that  this  form  of  A;^  is  just  the  same  as  if  it  had 
been  derived  from  A'  b}^  making  the  pith,  pgth,  •••^^th  rows  of 
A'  the  1st,  2d,  •••  A:th  rows,  making  the  same  changes  in  the  places 
of  the  ^ith,  goth,  ...  Qj^th  columns,  and  then  putting  1  for  each 
remaining  element  of  the  principal  diagonal,  and  0  for  every 
other  element  of  the  n  —  k  rows  of  which  A;^  is  not  a  part. 


Multiplying,  we  have 


AA, 

=    A     0   . 
0     A   . 


0    ttp^i  ...  Clp^q^-l    Ctpi^i+l  •••  (^p^q^-l   ^PiQi+l  '"  ^Pi^ 




• 



0     0 

...    A  ap^i.. 

•   ^PkQi-T^  "2^i-?i+l  • 

••  ^hk<li-'^  ^^PkQi+'i- 

0     0 

...   0   an  . 

•  «lgi-l     «l(Zi+l     • 

•.   «1(72-1     Cllq^+i 

0     0 

...  0   a2i   .. 

•  «2f?i-l     «2(7i+l     • 

"  «2(72-l     Ct2g,+1 

0      0     ...    0    a„i    ...  Clngi-l    CLnq^+l    •••  ^n^j-l    ^^nq^+l 


•••  Cipkn 

...  ain 

...  a-zn 

• . .  Cf/fin 


68  THEORY   OF   DETERMINANTS. 

Now  this  determinant  is  at  once  expressible  as  the  product 
of  two  determinate  factors,  and  we  have 

A\  —  A*  times  the  complementary  of  the  tninor  of  A  corre- 
sponding  to  A^  in  A'. 
Whence 

A;^  =  A*"-^  times  the  complementary  of  the  minor  of  A  corre- 
sponding to  A„of  A', 
as  was  to  be  shown. 


62.    From  the  preceding  article  it  follows  at  once  that  if 

A  =  0,  then 

••=0; 


An  Au 

-^21    -^22 

= 

= 

An  Aio 

Aqi    Aq2 

i.e.,  in  general 

whence 

Aij,      Ape 

0, 

or  A, J, :  Aie  ::Ap^:  Ap^. 

That  is  to  say  : 

If  A  =  0,  the  cof actors  of  the  elements  of  any  row  are  propor- 
tional to  the  cofactors  of  the  corresponding  elements  of  any  other 
row. 

From  the  preceding  article  we  have  also 

=  A  X  complementary  minor  of 


Aik    Ai. 


■^^pk       -^^pe 


which  may  be  written 

dA     dA 
dajj,   da/g 

dai^dap^ ' 

dA     dA 

dap,  dap. 

whence 

da 

1'A          dA    dA 
,,dape      da,,  da^. 

dA    dA 
dap,  da^. 

which  is  the  formula  already  obtained  in  57. 


GENERAL  PROPERTIES  OF  DETERMINANTS. 


69 


1.    Show  that 
0,     0 


EXAMPLES. 


also 


2.    Show  that 


0 
0 

^2 
^1 
0 

0 
0 

0 
0 

h 

h 
h 

\B, 


0 
0 

2/3 
2/2 

2/1 
0 

a4 

0 

0 

0 
0 


=  -2!i|a;i2/2%l    |2/i2^2|; 


|«1&4|       |0t2&5l       |«3&6 


C2 

0 

0 

^3 

0 

0 

ClA 

A 

A 

CiB, 

B, 

B, 

c,C\ 

c. 

c. 

=  \A^B^C^\    lai^gCs]. 


3.  If  A  is  a  determinant  of  the  nth  order,  having  n  —  m  zero 
elements  in  the  corresponding  places  of  m  rows,  then  A  is  the 
product  of  that  minor  whose  elements  are  the  other  elements  of 
the  771  rows  and  its  complementary  ;  the  sign  of  the  product  is 
determined  as  in  55. 

4.  If  any  determinant  of  the  nth  order  has  more  than  {n  —  m) 
zero  elements  in  the  corresponding  places  of  m  rows,  the  deter- 
minant vanishes. 


5. 


«! 

\ 

Cy 

M 

iV^ 

= 

a^-M 

h^-N 

0^2 

h 

C2 

P    Q 

a,-P 

h-Q 

0^3 

h 

C3 

ttg        63 

a^ 

h 

^'4 

a4    64 

as 

h 

C5 

as    65 

«! 

a^ 

ag 

a. 

=  \d,a,\    \h,c\ 

&i 

0 

0 

h 

Ci 

0 

0 

C4 

d. 

d. 

d. 

C^4 

03^4651; 


70 


THEORY   OF   DETERMINANTS. 


6. 

tti 

«2 

a. 

^4 

«5 

«6 

Oj     as 

ag 

«10 

h 

h 

h 

&4 

?>-, 

h 

h     h 

&9 

K 

0 

0 

0 

0 

0 

Ce 

Cr      Cg 

0 

0 

0 

0 

0 

0 

0 

f?6 

cZ;     dg 

0 

0 

0 

0 

0 

0 

0 

ec 

e?     eg 

0 

0 

/l 

/. 

f. 

/. 

/. 

./o 

/r     /s 

0 

0 

9i 

92 

9s 

i/4 

i/o 

^6 

i/z        ^8 

0 

0 

\ 

h 

h 

h 

h 

/'« 

/'r      /'8 

0 

0 

h 

h  . 

0 

0 

0 

?'(; 

^T          ^*8 

0 

0 

k. 

A:2 

0 

0 

0 

A'6 

Av     A-g 

0 

0 

= 

=  -| 

<-hK\   1^6 

CZyCg 

IIA 

74/^.1  \hK 

X 

7.   Show 

tlint 

{ay-h^y      (a.-boY 

(a,-b,r      {a,-b,r 
{a,-b,r      {a,-b.^' 


{a,-b,y 
{a,-b,y 
{a^-bsY 


K- 

-b.y 

(«.- 

-hy 

(«3- 

-b.y 

(a,- 

-hy 

=  0. 


This  may  be  proved  by  multiplying  the  two  arrays : 


and 


8.    Show  that 

|«ln|  (^1  +  3^2  + a^3  + 


+  ^.) 


(■("21         ^22 


+ 


an 

^21 


«12 


«nia^l   (^n2^2 


^2n 


^2» 

a„t,x,. 


+ 


^21  1  ^22*^2 


a2na;« 


Notice  that  the  coefficient  of  Xi  in  this  sum  is 

ay.Au  +  a2iA2i  +  a.^Asi  H h  a,,f^„i  =  |  ai„|. 

9.    As  an  application  of  the  preceding,  show  that 


2  (xi  +  a^a  +  a^s) 


1      1 


GENERAL  PROPERTIES  OF  DETERMINANTS. 


X,'  xi  xi  + 

X^    X^    X., 

+ 

Xo    X^    Xi 

x?  xi  x^- 

1   1   1 

1   1   1 

X;X.2  XoX^  X^Xi 

+ 

Xi       X,       X., 

X.J       X^       Xi 

XiX.2  X2X^  X^Xi 

1    1    1 

1     1     1 

1 0 .  Given  /^  (cc)  =  a^x"  +  3  b^x-  -f-  3  CiX  +  d^ , 
fo{x)  =  ao-Tr  4-  3  b.^oi^-i-  3  c.^  -j-  c?2, 
Mx)  =  apy'-\-  3  b.x'-^  3  c,x  +  d, ; 


show  that 


/iW     /I'W     /i"W  =-18 

^2^       f^'ix)       f,"{x) 

M^)     fs\x)     f,"{x) 


I    —X  x^  —  ar*^ 

h     bi  Ci     di 

I2     O2  C2     CI2 

h     ^3  ^3     ^3 

The  first  determinant  is  at  once  reducible  to 


-18 


ciiX  +  bi  biX  +  Ci  CiX-i-di 
agic-h^a  &2^  +  C2  C2X-\-d2 
b^x  +  C3     Cgic  +  da 


0-30;  + 63 
which  may  be  written 


10  0  0 

«!     a^x  +  bi     biX-\-Ci     CiOJ  +  c?! 

«3     a^x  +  b^     hx-\-c.i     c^x-\-d^ 

Again  using  37,  the  last  determinant  becomes  the  result  above 
written.  The  student's  attention  is  called  to  the  fact  that  the 
method  of  bordering  a  determinant,  z.e.,  increasing  its  degree 
without  changing  its  value,  here  employed,  is  frequently  of  use 
in  simplifying. 

63.  The  following  examples  comprise  several  interesting 
expansions  of  determinants.  The  cases  considered  and  the 
methods  employed  are  important. 

I.  Expand  the  following  determinant  in  ascending  powers 
of  xi 


X 


a 


72 


THEORY   OF  DETERMINANTS. 


A  is  evidently  a  function  of  x  of  the  nth  degree,  in  which  the 
coeflacient  of  x""  is  1,  and  the  absolute  term  is  /(0)  =  |ai„|. 
To  complete  the  expansion,  we  have  to  find  the  coefficient  of  a^. 
Consider  the  product  of  two  complementary  minors  of  A,  of 
the  feth  and  {n  —  k)  th  degrees  respectively, 


otee  +  a; 


a,,  +  x 


and 


%„-\-x 


This  product  contains  the  term 


x' 


=  ^n. 


say. 


The  entire  coefficient  of  ic*  is  accordingly  SDn-*,  *.e.,  the 
sum  of  all  the  minors  of  |ai„I  of  order  7i—k,  whose  principal 
diagonal  lies  in  the  principal  diagonal  of  lai„|. 

As  an  illustration,  the  student  may  show  that 
CLi-{-x       bi  Ci  di 

«3  ^3       C3  +  a;        ds 

a^  64  C4      d4-\-x 

+  \b2d^\-h\aM  +  \cM^x' 

For  another  exercise,  let  the  student  find  the  terms  of 
A  =  |ai„|  that  contain  k  elements  from  the  principal  diagonal,  by 
considering  the  product  of  two  complementary  minors,  as  above. 


GENERAL  PROPERTIES  OF  DETERMINANTS. 


73 


II.  Expand 

ax  +  ly        CiX  4-  n^y 

h^  -f  m^     a2X  +  ^22/ 
in  ascending  powers  of  x  and  y. 

Putting  first  y  =  0,  and  then  a; 
and  2/^,  respective!}*,  are 


\x  +  m^ 
a^x  4-  /i2/ 
ca;   -\-ny 


0,  the  terms  involving  a^ 


a     ci    &i 

,  and  f 

I     rii 

Wi 

Ca     h    a. 

?i2    m 

h 

&2      <^*2      C 

wig    ^2 

n 

Putting  the  y's  in  the  two  last  columns  of  A  equal  to  zero, 
we  obtain  for  one  set  of  terms  involving  afy 


3?y 


I 

Cl 

W2 

h 

m2 

<X2 

and  the  two  other  sets  of  terms  containing  x^y  are,  similarly, 
a?y 


a    Ui    bi 

and  a^y 

a    Ci    mi 

C2    m    ai 

C2    b     Zi 

&2     h     c 

&2      «2      ^ 

The  coefficient  of  xy^  is  found  in  a  similar  manner,  and  the 
entire  expansion  is  accordingly 


\=:X' 

a 

Ci    &1 

+  0^22/ 

I    c. 

M 

+ 

a   Ui 

&i 

+ 

a 

Ci  mi 

C2  b  a. 

712  6  tti 

C2  m  tti 

Ca    6    ^1 

62   ttg   c 

_ 

ma  Ota  C 

62    ^a  c 

62  ^3  n 

— 

4-a^ 

a  Wj  mj 
C2  m  ?! 

+ 

Z    Ci  mj 
W2   &    ?i 

+ 

?      Wi    &1 

rig  m   «! 

+  2/^ 

I   nj  mi 
na  ?>i    Zi 

_ 

^2     ?2 

n 

WI2 

^2    ^ 

m 

2   ?2     c 

ma  k 

n\ 

III.  Show  that  any  determinant  A  may  be  developed  in  terms 
of  the  elements  of  any  row  and  column  and  the  second  minors 
of  A  corresponding  to  the  product  of  these  elements. 

Let  A'=|rtii  ^22  agsh 

and  border  it  as  indicated  below ;  calling  the  result  A,  we  may 


74 


THEORY  OF    DETERMINANTS. 


expand  A  in  terms  of  the  bordering  elements  and  first  minors 
of  A',  i.e., 

A=  a^n  am  am  «„«  =  ctoo  A'—  J  aio  ttoi  Ai+«io  «02  ^12 

4-«10    <^03    ^13H-«20     «01     ^21+<^20     «02    -^22 
"i     a^     a^Q    -^23 "I    ^30     ^01     -^31T"  ^30     <^     -"32 

-f-  ago   ttos  -^^as  1 1 

in  which  Ai^^  is,  as  usual,  a  first  minor  (with  its  proper  sign) 
of  A'. 


ttoo 

«01 

«02 

^03 

«10 

«11 

«12 

ai3 

«20 

^21 

^22 

«23 

^30 

^31 

«S2 

^33 

In  general,  if  A'=  1  a^  022  •■ 


,  we  have 


aoo  ttoi  «02  ••• 

aon 

ft^Q  ail  %2  ••• 

au 

(7^0    ^21    «22    ••• 

«2« 

«nO«nl   «n2"- 

^*nn 

=  aoo A'—  2 afoaot^i*    (/, ^•  =  1, 2, 3  ...w) , 


in  which,  as  before,  Aij,  is  a  minor  of  A'. 

For  .the  terms  of  A  containing  a^  are  obviously  aooA'.     Now 
let  (7  be  the  complementary  niinor  of 

cioo  OoA    in  A  ; 

then  ttooCtiAC'  contains  all  the  terms  of  A  involving  Oooaf*;  hence 
afjO  contains  all  the  terms  of  A'  involving  (x,^,  and  consequently 

and  —  ai^aQj^Aij,  is  the  expression  for  the  terms  of  A  containing 
the  bordering  elements  a^o,  ao** 

This  expansion,  known  as  Cauchy's  Theorem,  is  frequently 
written 

A  ='«,,  A,-  :Sa,,a,,/?i,.  (a) 

Here  A  is  a  determinant  of  the  nth  order.  A^,  is,  as  usual,  the 
complementary  minor  of  a^,  in  A  ;  i  has  all  integral  values  from 
1  to  n,  except  r  ;  Tc  has  all  integral  values  from  1  to  w,  except  s  ; 
and  ^ik  is  the  complementary'  minor  of  a,j  in  A,,.  (a)  is, 
accordingly,  the  expansion  of  A  in  terms  of  the  elements  of  the 
rth  row  and  the  sth  column. 


GENERAL  PROPERTIES  OF  DETERMINANTS. 


75' 


The  student  may  show  that 


A  = 


A  = 


a  f  g  h 

A  b  0  0 

9i  ,0  c  0 

hj^  0  0  d 


ai-^Xi  ag       ag 


=  abed  —  Jficd  —  ggibd  —  hh^c. 


0 


a^ 

0 

0     —x^     Xq      0 
0       0      — ajg    x^ 


ai+iTi  ttg     ttg  a^ 

—  Xi     ct'a     0  0 

—  Xl  0  iCg  0 

0  a;^ 


-x"i     0      0 

f        ai     Og     ag     a4  "I 

=  XiXoXoX,  <  i-\ —  Y 

i     ^     J     4    j^         '    iCi         i»2         ^3         ^4  j 


IV.    If  A 


and  if  we  put 


Xl     ag    ttg    . 

..  a„ 

ai    X2     ttg    . 

. .  a„ 

tti   ag  ajg  . 

••    '^n 

«!     as     ttg    . 

..  a;„ 

f{x)  =  (x^—  ai)  (iK2-  02)  •••  (X,,-  aj, 


and 


aa;. 


we  find 
For 
A  = 


A=f(x)  +  :S,aJ'(x,). 


1 

0    0    0 

...    0 

1 

Xl     a.2     Og 

...    a„ 

1 

tti   a*2   ag 

•  •  •     Ct/t 

1 

ttj     Oo     iTg 

...     OLn 

1 

ttj  as  ag 

...   a;. 

1        fti  a2  ttg 

la.'i— ai      0  "         0 
1        0      X2-a2       0 

10  0  ajg—  ag 

1      0         0*         0 


0 
0 
0 

X^  —  a^ 


whence  (if,  as  in  III.,  we  let  A'  represent  the  complementar}' 
minor  of  the  first  element)  A'=/(a!),  and,  since  every  first 
minor  of  A'  vanishes  except  the  minors  of  the  diagonal  elements, 
we  have  the  required  value  of  A  on  applying  the  theorem 


76 


THEORY   OF   DETERMINANTS. 


V.    Show  that 


A  = 


CLi  >C  X  OS  , , ,  QC 
X  02  X  X  ...  X 
X    X    a^   X   ...    X 


=  f{x)-xf{x). 


X    X    X    X   ...    a„ 

in  which      f{x)  =  (x  —  aj)  {x  —  ag)  (a;  —  ag)  ...  (a;  —  a„) , 
and  f{x)  =  J^  =  {x-  a,,)  (a;  -  ag)  ...  (»  -  a„) 

+  {X-  ai)  {x  -  ag)  •••  (a;  -  a„) 

+  "'+(x-a,){x-a2)-"(x-  a„_i) . 

A=    10    0    0  ...   0    =    1    — a;    — x    — a; 

Itti— a;  0  0 
1  0  ag-a;  0 
10        0    ttg-a; 


1 

0 

0    0... 

0 

1 

ai 

X     X    ... 

X 

1 

X 

a2  X    ... 

X 

1 

X 

X      ttg  ... 

X 

1 

X 

X     X    ... 

a„ 

—  X 

0 
0 
0 

a,-X 


10        0        0 
Then,  as  in  the  preceding  example, 

A=/(a^)-<(a:). 

64.  To  the  expansions  of  the  preceding  article  we  append  the 
solutions  of  the  following  determinant  equations. 

I.    Solve  the  equation 


A  = 


X 

tti 

«! 

tti 

«! 

X 

ttl 

tti 

tti 

ai 

X 

ai 

«! 

tti 

tti 

X 

=  0. 


We  find  by  easy  reductions 


A=(a;-aiy 


X 

tti 

«! 

«! 

-1 

1 

0 

0 

-1 

0 

1 

0 

-1 

0 

0 

1 

=  (a;-ai)3(a;4-3ai)  =  0. 


Whence, 


a;  =  ai,  ai,  ttj,   —3  tti. 


GENERAL  PROPERTIES  OF  DETERMINANTS. 


77 


II.    Find  the  values  of  x  in  the  equation 


A  = 


X 

ai 

h 

C] 

ttl 

X 

Ci 

&1 

h 

Ci 

X 

a. 

Ci 

h 

«1 

X 

=  0. 


A  = 

a;+«i+^+Ci   «!   61   Ci 
ar+ai+6i+Ci   a;    Cj    61 

x-i-ai-\-bi-\-Ci   61   ai   a; 

=  (a;-f  ai+6i+Ci) 

1 
1 

1 

1 

X 

&1 

X    < 

^1 

=  (a;  +  ai+  &i+  Ci)  (a;  —  ai+  61—  Ci) 

0  -1     1 

1  X       Ci 

-1 

• 

1       Ci       X 

1     bi    «! 

a; 

Put  the  two  polynomial  factors  =A  and  J5  respectively 
e  last  expression 

then 

A.B. 

0  0-10 

1  x-hc^      Ci      61+ Ci 
1     x-\-Ci     X      ai-{-x 
1    6i-f-«i     «i     cii+a; 

= 
A.B. 

0  0 

1  0 
1              0 

1  feiH-Oi— a;  — ( 

-1 
a; 

0 

-ai- 

0 

0 

X 

• 

Whence 

{x  -\- «!+  61+  Cl)  (x  —  a^—  Ci+  &i)  (5i4-  ai— a;  —  Ci) 
(tti+aj  — &i— Ci)  =  0. 

.-.  a;  =  -(ai+6i+ci),   (a^- ftj-j-Ci),   (^^-Ci+ai), 
(61-tti+Ci). 

III.    Find  the  roots  of  the  equation 


A  = 


=  0. 


a^  W  (? 

{a-^xy     {b^-xy     {G+xy 
{2a  +  xy   {2b+xy   {2c-\-xy 

From  the  third  row  of  A  subtract  the  first  row  multiplied  by  8, 


78 


THEORY   OF  DETERMINANTS. 


aud  from  the  second  row  subtract  the  first  row.     Then  subtract 
the  second  row  from  the  third,  and  we  have 


=  3X2 


o?  V"  <? 

3a2  +  3aA.  +  A2     ZV^  +  ^hX  +  X^     Z<^  +  ZcX  +  X., 

3a2-f-aX  362  +  6X  Z(?-{-cX 


Now  subtract  the  third  row  from  the  second,  and 


A  =  3\'^ 


a^  h^  <? 

2a  +X       26-f-A.      2c+A 
Sa^H-aA     ?>W+hX     Zc'-^cX 


=  0. 


From  this  equation  it  is  obvious  that  three  values  of  X  are  zero ; 
the  other  two  roots  can  be  found  by  equating  to  zero  the  quad- 
ratic factor  of  the  first  number,  and  solving  for  A.. 

A  ma}^  however,  be  further  simplified  as  follows  :    subtract 
the  first  column  from  each  of  the  other  two  ;  then 


A==3\Hc-a)(6-(x) 


2a-\-X  2  2        ■ 

3tt2+aA.    36  +  3a+X     3c+3a4-A. 


Now  subtract  the  second  column  from  the  third,  and 


A=3\3(6-a)  {c-a)  (c-b) 


a?  a2-f-a6-f62     a-\-h  +  c 

2a+X  2  0 

3a2+a\     3a  +  364-X  3 


Finally,  add  the  second  column  multiplied  by  —a  and  the  third 
multiplied  by  ab  to  the  first,  and  afterward  subtract  the  third 
multipUed  by  a  +  &  from  the  second  ;  then 


A=3X«(6-a)  (c-a)  (c-6) 


ahc  —bc  —  ca  —  ab  a-\-b-\-c 
X  2  0 

0  A  3 


0. 


Whence  three  values  of  X  are  seen  to  be  zero,  aud  the  other 
two  roots  are  readilv  found  from  the  quadratic 


(a  +  6  +  c)  X'+  3  (&c  +  ac  +  a2>)  X  +  6  abc  =  0. 


GENERAL  PBOPERTIES  OF  DETERMINANTS. 


79 


65.  Theorem.  —  The  total  differential  of  a  determinant  A 
is  a  sum  of  n  determinants,  each  oftvhich  is  obtained  from  A  by 
substituting  the  differeyitials  of  the  elements  of  a  row  for  the 
elements  themselves. 

Let                              A=  \x^y2Z^---t^\. 
Developing  in  terms  of  the  elements  of  the  ith.  row, 
A=    x,X,-\-   y,Y,-\-   ;2^z,+  ...  +    t.T^. 
.-.  dA  =  dx,Xi+dyiYi+dZiZi-\ \-dt,Ti. 

There  must  be  n  such  expressions  for  the  total  differential,  each 
of  which  is  obviously  A,  after  changing  the  elements  of  the  ^th 
row  into  their  differentials. 

.-.  [c2A]*  = 


+ 


From  the  differentials,  partial  or  total,  we,  of  course,  pass 
to  the  coiTesponding  derivatives  in  the  usual  way. 

Illustrations. 


dxi  dyi  dzy 

^2  y-i  % 

...dt, 

...     t2 

+ 

»i    2/1    % 
dx2  dy2  dz2 

...  t, 

,..dt2 

^n      Vn      ^n 

...     t,, 

^u     Vn      ^n 

...  t. 

X^      2/1      ^1 
^2      2/2      ^2 

...     t, 
...      t2. 

• 

r 

dx^  dy^  dz^ 

'.'.'.   di 

V  —  —  ;  N'^dv 

N 


dM  M 

dN  N 

.      d 

dM  M 

dN  N 

= 

d'^M  M 

d'jsr  N 

Let 


(^U  ^22  ^33  ^44 1 


a  26 

c    0 

0    a 

26  c 

b  2c 

k    0 

0    b 

2c  k 

dA 

da 


J^ll-\-A22  — 


a    26   c 

+  Jc 

2c    k    0 

6    2c   k 

a   c 
6    k 


The  [  ]  denote  the  total  differential. 


80 


THEORY   OF   DETERMINANTS. 


^  =  2A^+2A^+A,,-^A^=4b\l    k 


-4k 


a   b 
b   c 


2b 
a 
b 


c 

26 
2c 


a    c 
b    k 


■~7~  — -^13"f'-^24~f~  2As2-}-  2A4Q  =  •••, 

ac 


dA 
dk 
66.    Theorem 


-^33  +  -^44 


dA 


If  the  elements  of  A  are  all  functions  of  the 

same  variable  x,  ^^—  equals  the  sum  of  n  determinants,  each  of 
ux 

which  is  obtained  from  A  by  substituting  the  derivatives  of  the 
elements  of  a  row  for  the  elements  themselves. 

The  truth  of  this  proposition  is  evident  from  the  preceding. 

Thus,  if 


6A 
dx 


Mx)    f22ix)   . 
/nl(aj)  fn2{x)   . 

/2l(a5)     /22(«) 


f2n{x) 
fnn(x) 

JJix) 

•  Mx) 

'fnnix) 


+ 


/uW   fu{x)  .../in(a;) 
fj(^\   f.J(r.\      fj(^a:) 


+ 


If 


1     Xiic     0  0 

1       1      Aaic  0 

0       1        1  X^x 

1  1 


0      0 


fn'ix)  f^\x) 

fnl{x)     fAx) 

fn(x)    f^(x) 
U{x)   f^x) 

fnl'(x)  fj(x) 
=  A1A.2X3      1 


-fnnix) 

'An{x) 
'f2n{x) 

Jnn'ix) 


+  ••• 


-      a;      0     0 
X     0 


1  1_ 
A2  A2 
1      1 

0     0       1 


0 


—     X 
A,      ^ 


GENERAL  PROPERTIES  OF  DETERMINANTS. 


81 


the  student  may  show  that 


dx 


—  —  AiX2^S 


1 


X 


^3 

0     1     1 


+ 


—     a;  0 

0     ^  X 

A3 

0     0  1 


+ 


1 

X 

0 

Ai 

1 

1 

X 

A2 

A2 

0 

0 

1 

"b 


CHAPTER  III. 

APPLICATIONS   AND   SPECIAL   FORMS. 

67.  We  have  now  discussed  the  origin  and  some  of  the 
properties  of  determinants  ;  it  remains  to  show  how  useful  these 
functions  are  in  application,  and  to  examine  some  of  the  Special 
Forms  that  are  of  frequent  occurrence.  Within  the  limits  of  an 
elementary  work  like  this  it  will  be  possible  to  select  only  a 
very  few  of  the  many  important  applications,  and  to  touch 
somewhat  briefly  upon  the  special  forms.  Enough  will  be  given, 
however,  to  enable  the  student  to  pursue  his  further  investiga- 
tions with  pleasure  and  profit.  We  now  return  to  the  problem 
with  which  we  commenced  the  presentation  of  determinants,  and 
proceed  to  the 

^^  Solution  of  Linear  Equations,  and  Elimination. 

68.  Consider  the  set  of  three  simultaneous  linear  equations  : 

aiX  +  6i2/  +  CiZ  =  7?ii 


«1 

h 

Ci 

a^ 

h 

C2 

cts 

h 

C3 

a<fc  +  b^y  -\-  c^  =  ^2  [>■ ,    and  A 
a^x -i- b^ -^  c^z  =  7ns 

Multiply  these  equations  by  Ai,  A^,  and  A^  respectively,  and 
add  by  columns,  obtaining  : 

(aiAi+  a2A2+asAs)x  +  (61^1+62^4-^3^3)2/ 
+  (Ci^l  +  C2A +03^3)2 
=  miAi-\-m2A2-\-m^As. 

By  45  the  coefficients  of  y  and  z  vanish  ;  the  coefficient  of  x 

is  A  =  |ai62C3l,  and  the  absolute  term  is  I  mi  62  C3 1. 

Whence  ^,_  \m^  b^  Cgl^ 

ki  62  C3I 


APPLICATIONS   AND   SPECIAL   FORMS. 


83 


y  = 


If  we  had  multiplied  the  given  equations  by  By^  -Bg,  -B3,  we 
should  have  caused  the  coefficients  of  x  and  z  to  disappear  in 
the  resulting  equation,  and  would  have  found 

1^1  ^2    C3I 

Using  (7i,  C2,  C3  as  multipliers,  we  should  find,  similarly, 

^_  ki  62  ^3!, 
ki  62  C3! 

69.  To  generalize  the  solution  of  the  preceding  article  is  now 
an  easy  step.     Given 


and 


^21^1  "I     tt22*^2  "I     •  *  *  "T"  tt2r'^r     I 
a,^X^  4-  «V2^2  H f-  (^rrXr  + 


■'  +  a2nX^  =  m,2 

■  *  "T"  Clrn'^n  =  '^r 


,     I., 


«11 

«12 

...    ai,    . 

•      «ln 

«21 

^22 

..    ttgr    . 

.      «2n 

a,i 

a,2 

..    a,r    . 

.      «,.« 

am 

«„2 

..      ttnr     . 

.   a„„ 

Here  A  is,  as  before,  the  determinant  formed  from  the  n^  co- 
efficients in  the  first  members  of  equations  I.,  and  is  called  the 
determinant  of  the  system. 

Multiplying  equations  I.  in  order  by  A^^^  A^r-,  ...  A„^  ...  yl„^, 
and  adding  by  columns,  we  find 

+  OjriArr  +  *  *  *  +  f^nzAnr)  ^2 


I    \^12Air  -f-  (^22A2r  ~\~  '  ' 
+  ... 

+  


+  a„,^,rH f-  a„,,A^,)x^ 

-\-m,A„-\ Vm^A^,. 


(^) 


84 


THEORY   OF  DETERMINANTS. 


In  equation  (A)  the  coefficient  of  all  the  unknowns  except  the 
coefficient  of  x^  vanish,  and  the  coefficient  of  x^  is  obviously  A. 
The  second  member  of  (A)  is  evidently  what  A  becomes  when 
mi,  mg,  ...m^  are  put  for  the  corresponding  elements  of  the  rth 
column.     Hence 


ttll 

a,2 

mi 

...    tti. 

^21 

^22 

Wg 

...      ^2. 

ftrl 

a.2 

m. 

...    a,„ 

am 

(^n2 

w« 

...   a„„ 

ttn    ai2    .. 

.     «!,     . 

.    a,. 

^21      ^22      . . 

aa,    . 

.    a2« 

-^ 

a,.i   ay2   .. 

arr     . 

.    a^ 

a„i  a„2   .. 

a„,   . 

.   a«n 

Translating  this  formula,  we  have : 

The  value  of  each  of  n  unknowns  in  a  set  of  n  linear  simul- 
taneous equations  is  the  quotient  of  two  determinants;  the  divisor^ 
(denominator)  is  the  same  for  all  the  unknowns  and  is  the  deter- 
minant A  of  the  7ith  degree  formed  by  writing  the  coefficients  of 
the  unknowns  in  order  (i.e.,  the  determinant  of  the  system)  ;  the 
numerator  of  the  value  of  any  unknown  as  x^  is  obtained  from  A 
by  substituting  for  the  elements  of  its  rth  column  the  second  mem- 
bers of  the  given  equations  in  order.* 

70.  The  following  modification  of  tho  solution  already  given 
of  equations  I.  will  be  interesting.  Employing  the  same  notation 
as  in  69,  we  have 


flj^A 


which,  by  37, 


an 

a^2   • 

^21 

a^   . 

... 

...    . 

a,i 

a.2   . 

a„i 

a«2    • 

airXr 

a^rXf 

... 

a^x. 

... 

a«r»r 

«1« 

a«« 


*  This  is  the  rule  for  the  solution  of  simultaneous  linear  equations  first 
obtained  by  Leibnitz,  and  subsequently  rediscovered  by  Cramer.  (See 
opening  paragraph  of  Chapter  I.) 


APPLICATIONS  AND   SPECIAL   FORMS. 


85 


^2r-l        ^21'^l~r  ^22*^2    1" 
-f-  a2n  ^»       C(-2r+i 


•  •  "f"  Cllr-l^r-l'T'  (^Ir-^rl 
"  +  Ot2r-l^r-l~H  Cl2r^r~\~ 

"  +  arr-lX,_i-^  a,rXr  + 

"  +  anr-lXr-i+ar,rXr-\- 

<^2/i 


Now  substitute  in  the   last  determinant  the  values   of  the 
elements  of  the  rth  column,  and 


a?  A  = 


Oil    ai2   . . . 

^21       ^22      •  • ' 


Ot,,i       Cl„ 


«2r-l 


«1« 
a2n 


a,,_i    m. 


m. 


..  a„ 


I  Cll\(^22  •  •  •  ^rr  •  •  •  ^nn  ' 

as  before. 

A  simple  example  of  the  methods  of  69  and  70  is  the  solution 
of  the  following  equations  : 


.'.  X 


2x  +  6y  —  3z=lS 
Sx-Sy-\-2z  =  21 

48  3  3 
18  6-3 
21-3     2 


Here  A  = 


5  3  3 
2  6-3 
8-3     2 


=  -231, 


-231 


3;  2/  = 


5     48 

3 

2     18- 

-3 

8     21 

2 

-231 


5:  z= 


5  3  48 
2  6  18 
8-3     21 


231 


=  6. 


As  another  example,  we  may  solve  the  equations : 


z  H-w  +flj  =  b 
ti-{-x  -\-y  =  c 
^  -\-y  +2J  =d 


Here  A  = 


0 

1 

1 

1 

1 

0 

1 

1 

1 

1 

0 

1 

1 

1 

1 

0 

=  -3. 


86 


THEORY   OF  DETERMINANTS. 


The  student  may  show  that 

a;  =  i(6+c-f  d-2a);     2/ =  J(c +d  +  a- 26)  ; 
z=l(d-{-a-\-b-2c);     u  =  l{a-\'b -^c  -2d). 

71.  We  have  hitherto  tacitly  assumed  that  neither  A  nor  m< 
(i=  1,  2,  ...  7j)  should  vanish.  If  A  vanishes  and  m^  does  not, 
the  value  of  each  unknown  becomes  infinite.  If  m^  vanishes 
while  A  does  not,  the  values  of  the  unknowns  are  severally  zero  ; 
but  when  mj  vanishes,  the  system  consists  of  homogeneous 
equations,  and  their  solution  is  given  later.  If  m^  does  not  van- 
ish, but  A  and  the  numerators  of  the  unknowns  do  vanish,  then 
we  have  the  following  theorem. 

72.  IftJie  equations  of  a  set  are  not  independant,  i.e.,  if  any 
one  {or  more)  is  a  consequence  of  the  others,  the  value  of  each 

imknown  takes  the  form  — 


Since  the  equations  are  all  linear,  any  one  can  be  derived  from 
the  others  only  by  the  addition  of  two  or  more  of  them  after 
each  has  been  multiplied  by  some  constant  factor.  But  this 
gives  rise  in  the  determinant  numerator  and  denominator  of  the 
value  of  any  unknown  to  two  or  more  identical  rows,  and  hence 
numerator  and  denominator  vanish. 

For  an  example,  take 


aiXi-[-biX2    +CiXs    =  mi 
OoCCj  +  62^2     +  C2a;3     =  '^h 
aiXi-\-  aibiX2+  aiC^x^^  aimi 
We  find 

h  &i  Ci 


where  A  =  a. 


Xi=  tti 


^2  &2  ^2 

mi  61  Ci 


=  5;   a.,  =  ai'^^-^  =  5;   x,^ 
0  A  0 


«!  61  Ci 

ao  &2  ^2 
ttj  bi  Ci 


\ai  &2^%I 


0. 


For  a  second  example,  the  student  may  show  that  the  values 
of  the  unknowns  in  the  following  equations  take  the  form  -• 

4 


3a;  +  2?/  — 02  = 
6a;  — 3?/ -f-  4z 
y-2z 


=     4^ 

=   22  L 
=  -2J 


APPLICATIONS   AND    SPECIAL   FORMS.  87 

m^_i  =  0,  and  one  m  as  m^  does  not, 


^n^nn 


73.  If  mi  =  W2==  •• 
we  evidently  get 

_  m^A,a 

Whence       J^  =  ^=  ... -|!L  =  !??l\ 

74.  If   mi  =  m2=  •••  =m„=0,   ^.e.,   i/*  equations  I.  become 
homogeneous,  tJien,  unless  x^,  x^^  •••^«  «»*e  severally  zero,  A  mi^s^ 

In  that  case,  equations  I.  become 


?!  =  <Xn^i+  tti2CC2  + 


+  a2^ir,.+ 


+  «2Ha^n=0 


0  (m^  being  zero),  the 


II. 


l^  =  arl^l+  «r2^2H h  «rr^%  + 

Since   a:^A  =  I  an  a22  <^33  •  •  •  'wi,.  •••  <^nnl 
truth  of  the  assertion  is  obvious. 

An  example  is  furnished  by  the  homogeneous  equations : 

an  a^i -f  ai2  if2  4- «i3  a?3  =  0  ^ 

a2ia?l  +  «22^2  +  «23»^3-=0     I.  {E) 

ctsiX^  +  a32a^2  +  «33a^3  =  0  J 

Multiplying  equations  (jEJ)  by  A^,  A^i,  ^31,  respectively,  and 
adding  by  columns,  we  have 

(an^n  +  «2i^2i  +  «3i^3i)a^i 

+  («12^11  +  «22^21  +  a32^3l)^2 

4-  (ai3^ii  +  a-23^21  +  033^31)  a's  =  0. 
The  coefficients  of  X2  and  x^  are  zero,  and  we  have 

a;iA  =  0,  .-.  A  =  0. 

As  a  further  illustration,  the  student  may  show  that  if 

nXiX-\-vyiy+icz^z-\-Ui{yz^+yiz)  -i-Vj^izXi-^z^x)  -\-w^{xy^+x^y) 
is  zero  for  all  values  of  x,  y,  and  z,  then 

uvw  —  uui—  vv^—  wWi-\-  2  ?/ii'iWi=  0. 


88 


THEOKY   OF   DETERMINANTS. 


Observe  that  by  the  given  conditions  the  coefficients  of  a;, 
?/,  z  must  severally  vanish. 

75.  With  the  help  of  74  we  obtain  an  interesting  proof  of 
the  multiplication  theorem  of  53.  Consider  the  simultaneous 
equations 

(ai-X)a;i-t-       h^x^      +      qajg      =0^ 
as^i       +(&2  — A)a72+      Cg^s      =0  >  I. 
(h^      +      h^2      +(C3-X)a;3=0  ) 


By  the  preceding  article  we  must  have 


«!  —  A       6i  Ci 

CI2        62  —  •^      C2 

«3  &3         Cg  —  X 


or 


=  0, 


0; 


(a) 


X^-Jfx^  +  iVA-P 
where  we  notice  especially  that 

P  =   I  «!    62    C3  I  . 

Let  the  roots  of  (a)  be  Ai,  Ag,  A3 ;  then,  evidently, 

P  =  —  Ai  A2  A3. 

Now,  from  I.  we  obtain  three  new  equations  as  follows : 
Multiply  equations  I.  b}^  aj,  as,  ag  respectively,  and  add  them 
together ;  also  multiply  equations  I.  by  ft,  ft,  ft  respectively, 
and  add ;  finally,  multiply  equations  I.  by  yi,  yg?  73  respec- 
tively, and  add.  We  now  have  three  new  equations  where  the 
determinant  of  the  S3'stem  is 

A'=   aiai  +  «2a2  +  C^3a3  — ttiA       ftittj  +  &2a2  +  ?>3a3  —  a^A 

C^lft+  «2ft+  <X3ft-  ftA         61ft  +  62ft  +  ?>3ft-  ftA 

«i7i  +  0^2  72  +  «3  73  —  71^      &i7i  +  ^272  +  &3  73  —  72^ 


Cittj  -}-  C2a2  4-  Cgag  —  ttgA 

Cift+C2ftH-C3ft-ftA 
Ci7i  +  C272  +  C3y3  — ygA 


0, 


or 


QA3-JIfiA2  +  iViA-Pi  =  0, 


(&) 


APPLICATIONS   AND   SPECIAL  FORMS. 


89 


where  we  observe  that  Pi  is  what  A'  becomes  when  we  put 
X  =  0,  and  that 

Q  =  lai  ^2  ysl* 
Further,  since 


it  follows  that 


—  —  —  A.1   Ag   Ag  —  P, 

Pi  =  PQ=  Ui  &2  C3I  X  Ui  ^2  ysl- 


But  Pi  is  exactly  the  determinant  obtained  by  53,  and  this 
was  to  be  shown. 


76.  The  condition  A  =  0  being  fulfilled,  the  equations  no 
longer  determine  the  actual  values  of  the  unknowns  ;  they  deter- 
mine only  the  ratios  of  these  values.  For,  if  a;/,  x^-,  ...xj 
satisfy  equations  II.,  so  will  Jcxi,  kx^j...  'kxj,  k  being  any  factor. 
Any  n  — 1  of  the  given  equations  will  suffice  in  general  to  de- 
termine the  ratios  of  n~l  of  the  unknowns  to  the  remaining 
one.  An  example  will  make  this  clear.  We  employ  for  brevity 
only  three  equations : 


aiX-{-biy  -|-Ci2;  =  0 
a2X  -\-b2y  -{-C2Z  =  0 
a^x -{- b^y  +  CqZ  =  0 

Write  these  equations 

X 


(a), 


tti  -  4-  Ci  - 

y       y 


b. 


X  z 


[if>)' 


X  .       z 

«3-+C3- 


-b. 


From  any  two  of  equations  (b)  we  may  find  the  values  of 


X     z 

-  ;   thus  from  the  first  two 


y   y 


\bi  Cgl  .  z 

1%  Col '  y 


\cti  &2I 
|ai  C2I 


90  THEORY   OF  DETERMINANTS. 


Again,  equations  (6)  are   to  be  simultaneous ;  hence  these 
lues  of  -  an( 

y 

Substituting, 


dC  z  '     ' 

values  of  -  and  -  must  satisf}^  the  third  equation 


ttgl&i  C2I  —  ftglai  Col  +  Cgltti  62I  =  0  ; 
or  A  =  0. 

Since  from  the  preceding  equations  -  also  equals 

y 

l&i  C3I  I&2   C3I 


and  hence 


or, 


Idi  Csl  las  C3I 

we  have  ^  —  Al  —  -^  —  -^ 

y"  Br  B~  b; 
In  the  same  way, 

z  _  C^  _  C2  __  C3 

y    B,    B,    b; 

z      Oi      C2      O3 

x:y:z  =  Ai:Bi:Ci 
=  ^2  •  B2 '  G2 
=  ^3:^3:03.  ' 

That  is  to  say.  The  ratio  of  any  two  unknowns  in  a  set  of 
homogeneous  equations  is  equal  to  the  ratio  of  the  cofactors  in  A 
of  the  coefficients  of  these  unknowns  in  any  of  the  given  equations. 

The  general  proof  of  the  proposition  just  stated  may  be  given 
as  follows.     "We  have  to  show  (equations  II.)  that 

Xi'.x^'.Xs:  ••• :  x^ :  .•• :  x^^Au  :  ^12 :  -i4i3 :  .•• :  ^1, :  ••• :  A^^ 
=  A21 :  -^22 :  A2S  :  '"  :  A^r :  •  •  •  :  Am 

^=^  A^\\An2''A^'.  "*'.  A^y.  *.'"'.  Ann* 

If  these  proportions  are  true,  we  must  have  the  equations 
x^=\Aj^  (X  =  constant;  p  =  1,  2,  •••  n).  (^1) 


APPLICATIONS   AND   SPECIAL  FORMS.  91 

The  equation  ^ 

is  always  true,  whatever  the  value  of  p,  since  A  is  itself  zero. 
Substituting  in  (^2)  the  values  of 

as  obtained  from  (J5Ji),  and  multiplying  by  X,  there  results 
a.iXi  -f  a,2a;2  +  a.3^3  H h  (^rr^r  H h  ^rna^n  =  0. 

This  last  being  a  true  equation,  the  proportions  from  which  it  is 
derived  must  hold.* 

/■.-/^'• 

77.  From  the  last  article,  or  the  two  preceding  articles, 
we  deduce  the  important  conclusion.  In  order  that  n  linear 
homogeneous  equations  may  be  simultaneous,  it  is  necessary  and 
sufficient  that  the  determinant  of  the  system  vanishes.  In  that 
case  any  one  of  the  equations  is  expressible  linearly  in  terms  of 
all  the  others,  provided  the  first  minors  A^^  do  not  all  vanish. 
For  we  have  in  general,  A  being  zero,  and  Zi,  Zg,  ...Z„  repre- 
senting the  linear  functions  of  equations  II., 

hAiie  +  hAzk  +  •  •  •  +  ^nAnk  =  0  ; 

hence,  if  one  at  least  of  the  first  minors  Au,,  A2U, . . .  A^^  is  not  zero, 
as  for  example  A^^,  l^  must  be  expressible  linearly  in  terms  of 
/g,  ?3,  '"In-)  and  hence  Zi  =  0  is  superfluous.  If  all  the  first 
minors  vanish,  and  one  at  least  of  the  second  minors  does  not, 
then,  similarly,  it  may  be  shown  that  two  equations  are  super- 
fluous, the  system  being  doubly  indeterminate,  and  so  on. 

78.  Among  the  proportions  of  article  76  consider  the 
following : 

Xi'.  X2'.  x^:  •••  Xn  =-4„i :  An2  :  A^  :  •••  Anr^.  (P) 

*  This  demonstration  applies  of  course  so  long  as  tlie  first  minors  of  A 
do  not  all  vanish.  V 


92  THEORY   OF   DETERMINANTS. 

^ni5  ^«2i  ^«3i  '•'^nn   ^16,  nooe  of   them,   functions  of   the 
coefficients  of  the  last  equation  of  set  II.  in  74, 

Hence,    proportions    (P)  give  the   ratios  of  the  unknowns 

ajj,  iCg,  a^a,  •••  a;„,  that  satisfy  the  n  — 1  equations 

a^iXi    -{-ai2X2    -i \-auX^    ^^1 

a2iXi    -\-a22X2    -\ \-a2„x,,     =0    liii.^ 

if  we  denote  by  A^i  the  determinant  formed  from  the  co- 
efficients in  equations  III.  after  suppressing  the  first  column  of 
terms,  by  ^„2  the  determinant  formed  from  the  coefficients  of 
equations  III.  after  suppressing  the  second  cohimn  of  terms, 
and  so  on.  Hence  having  given  n  homogeneous  equations  con- 
taining n  + 1  unknowns 

anXj^-\-ai2X2-\ l-ai«+i«n+i  =  0  "^ 

a2ii»i  4-  «22^2  -\ f-  (^2n+lXn+l  =  0       I  jy^^        " 

aniXi-{-a„2X2-] h«nn+ia^'«+i  =  0   J 

we  find  the  ratios  of  the  unknowns  as  follows : 


put  A,  =  (-iy 


(Xjl     ai2     '•'      «i,_i     CLii-^i     •••      «l»-fi 
^21      ^22      *'•      ^2J-1      ^'2i+l      *••      ^2n+l 


Then  from  what  precedes 

x^:  X2:  x^:  '"  :  x^T^  =  di,:  ^2'  ^3'  '"  '  A«+i- 

79.    Consider   the   following  n  equations   containing  w  —  1 
unknowns. 

aiiXi    +ai2X2     H hoT'in-i^'i        +i>i    =0 

«2ia^l       +a22a?2        H |-«2n-iaJn-l       +i?2       =0 

dn-llXl-h  O.n-1^2  H h  a«-ln-iaJn-l  +  Pn-l=  0 


APPLICATIONS  AND   SPECIAL  FORMS. 


93 


Equations  V.  may  be  made  homogeneous  by  multiplying  them 
by  u,  and  regarding  XiU,  X2U,  ...x^u^  u,  as  the  unknowns,  u 
being  any  arbitrary  quantity.  Whence,  if  these  equations  are 
simultaneous,  we  have  by  77 

=  0. 


On 

ai2     ...    am-i 

Pi 

^21 

a22     ...    a2n-i 

P2 

... 

...    ... 

... 

CTn-n 

Gt'n-12  . . .      Ctn-ln-1 

Pn- 

am 

a«2       ...      (^nn-1 

Pn 

This  result  may  be  expressed  as  follows :  n  equations  (not 
homogeneous)  containing  n  —  1  unknoicns  are  simultaneous  if  the 
determinant  of  the  nth  degree  formed  from  alt  the  coefficients 
(the  second  members  of  the  equations  being  included  among  these 
coefficients)  vanishes. 

This  condition  could  also  be  derived  from  equations  II.,  Art. 
74,  by  putting  ic^=l.  Those  equations,  n  in  number,  then 
contain  n  —  1  unknowns  ;  and  if  the  equations  are  simultaneous, 
we  see  that  lai^l  must  vanish. 

80.  With  the  help  of  the  preceding  article  another  solution 
of  a  set  of  linear  equations  may  be  obtained.  For  brevity  we 
employ  only  three  equations  : 

(1)   aiXi-{-biX2-^CiXs=mi 

(Z)  a2Xi  -\-  O2X2  -J-  e2X^ 

(o)  a^Xi  +  b'iX2  -\-  CqXq  —  iivQ. 

Take  with  these  equations  another, 

(4)  o^ Xi  -\-b^X2-\-  C4 Xs  =  m^y 

which  we  suppose  consistent  with  the  first  three,  and  in  which 
^^4?  &49  C4,  m^  are  undetermined.     By  79 


=  ma  L 
=  mnJ 


or, 


where,  as  usual, 


\ai  62  C3  m^\  =  0; 
a^A^  -f  64 ^4  -f-  cJJ^  +  W4 Jf4  =  0  ; 

A^  =  —  \bi  C2  mgl  ;    ^4  =  I aj  Cg  mgl  ; 
C4  =  — !«!  62^31  ;    Jf4=Iai  62^3!  =  ^. 


(5) 


94  THEORY   OF   DETERMINANTS. 

Now  if  we  eliminate  m^  from  equations  (4)  and  (5),  we  get 

Since  equation  (6)  must  be  true  whatever  the  values  of  a^,  64,  C4, 
the  coefficients  of  a^,  64,  c^  severally  vanish. 


A, 


fl?2 


__:?4. 


or, 


_  \mi  62  C3I  .        _  \ai  mo  cj 

Xi  — -^     ■  ;    Xo  —         -= 

A  A 


O4. 

"aT  ' 


81.  Let  us  now  return  to  equations  I.  Art.  69.  Considering 
mi,  ma,  mg,  ...  m^  as  linear  functions  of  the  ic's,  we  can  express 
any  new  linear  function 

CiXi  +  C2X2  H h  c,ifl;„  =  2/ 

in  terms  of  the  m's. 
Thus,  if  we  have  given 


by  79, 


aiiXi-}-ai2X2  + 
a2ii»i4-a22^*2  + 

A'  = 


•  -i-CnXr,      =2/        ^ 

•  +  CllnXn    =  Wi      I 


or 


Ci    C2     ...  c„    2/ 
«!!  ai2  ...  «!„  mi 

a2i    a22     ...    Cl2n    ^2 


a„i  a„3  ...  a„„  m„ 
Now  if  A  =  Itti,,! ,  we  readily  obtain 
A'±2/A  =  ±2/A; 
Ci    C2     ...     c„    0 


+  a,.«a;n  =  m„ 
=  0. 


±2/A  = 


an  aj2 

^21    <^22 


a2„  m2 
a„„  m„ 


APPLICATIONS   AND   SPECIAL   FORMS. 


96 


82.  We  have  seen  that  if  7i  homogeneous  equations  are  to  be 
consistent  with  each  other  (simultaneous),  the  determinant  of 
the  system  must  vanish.     The  equation 

A  =  0 

then  is  an  equation  of  relation  between  the  coefficients,  and  is 
really  the  result  of  eliminating  the  unknowns  from  the  given 
equations.  We  shall  soon  investigate  this  resulting  equation  of 
condition  or  resultant  in  detail.  We  here  deduce  a  general 
form  by  which  the  result  of  eliminating  n  unknowns  from  p 
given  Imear  equations,  supposed  simultaneous,  may  be  ex- 
pressed, 2^  being  greater  than  n. 
Given 


VII, 


(hi^i  -{-a^X2-\-" 

. +ai«a;„  =0 

a2iXi-i-a.22X2-{-" 

+  «2n^«    =  0 

«nia^i  +  a„2«^2+" 

.+a„„a;„=0 

aj,iXi  +  aj,2X2  +  ...  +  Sn«n  =  0 


If  these  equations  are  to  be  satisfied  for  other  than  zero  val- 
ues of  the  variables,  the  determinant  of  the  system  for  any  n 
of  them  must  vanish  by  77.  The  equation  expressing  this  con- 
dition is  obtained  by  writing 


an  .  ai2 

^^21       <^22 


0. 


(M) 


Equation  (Jf )  is  accordingly  interpreted  to  mean  that  every 
determinant  of  the  nth  order  formed  from  any  n  rows  of  the 
matrix  on  the  left  must  vanish.  For  an  example  the  student 
may  eliminate  the  two  ratios  a?! :  iCg :  x^  from  the  five  equations 

aiXi  +  biX2-\-CiX^  =  0   (i=  1,  2,...5), 


96 


THEOEY   OF  DETERMINANTS. 


obtaining  the  equation 


«! 

h 

Cl 

^2 

h 

C2 

as 

h 

C3 

a^ 

h 

C4 

as 


0,  or 


83.    Suppose  we  have  given 


ttl 

as 

ttg 

a4 

«5 

&1 

&2 

2>3 

&4 

h 

Ci 

C2 

C3 

C4 

Cs 

then,  by  78, 

c^Xs  +  d^x^^ 
C2CC3  +  d^^ 
c^Xs  +  d^^ 

=  0, 
=  0, 
=  0; 

Xi'.x^'.x^: 

i»4=l^i  C2  ds\  :  — 

laj^c^dsl  : 

kiJ 

Substituting 

the  values  of 

x^      x^ 

■ — •)     — "i 

Xs          Xs 

i»4 

— ? 

l«i  h  dsl  '  —1%  62C3I 


we  get  the  relations 


61 1  ai  C2  c?3 1  +  Ci  I  «!  &2  c?3 1  —  (^1 1  ai  62 


«i  1  ^1  C2  ds 

a2  1  61  C2  C^3  1   —  &2  1  <^l  C2  C?3  1   +  C2  I  «!  &2  <^3  I 
«3  I  &1  C2  C^s  1   —  &3  1  tti  C2  ^3  1   +  C3  1  tti  62  C^3  I 

which  are  all  expressed  by  the  matrix 


0, 


C?2  I  tti  62  C3  1  =  0, 

ds  I  »!  62  C3 1  =  0, 


ai 

h 

Ci 

^1 

a^ 

h 

C2 

d2 

as 

h 

Cs 

ds 

To  generalize  this,  we  return  to  art.  78. 
From  equations  IV.  we  found 

x^'.x^iXs'.'"  a;„+i  =  Ai :  Ag :  A3  ••.  A„+i. 

Substituting  in  equations  IV.  the  values  of 


— •> 

X, 


Wi 


APPLICATIONS   AND   SPECIAL  FORMS, 
given  by  these  proportions,  we  have 

ttiiAi  4-  aio^a  -\ f-  (^hr  X-\ f-  «ln+l  ^«+i  —  0    1 

ttsiAi  +  a.22^2  H 1-  «'2,-  A,  H h  cf2H+i  A^+i  =  0 

a.iAi  +  a,2A2  H f-  a,.  A,  H \-  a,„+i  A„^i  =  0 

a„iAi  -f  a„2A2  H h  «»rA,  H f-  a^n+An+i  =  0    J 

These  n  relations  are  expressed  by  the  matrix 


97 


(li) 


W'21        ^22 

an     a,2 


«2r 


ttln 

«ln+l 

a2« 

«2«+l 

... 

... 

«,-n 

ttm+l 

... 

... 

«nn 

«nn+l 

Jf. 


We  have  accordingl}',  in  general,  from  a  matrix  of  the  form 
M,  the  following  relations  ; 


Ctrl 


Ou 

«13 

...  Oi, 

•••     «ln     ttln+l 

Ot22 

a23 

•••  a^r 

...    a2n    «2n+l 

a.2 

a,3 

...  a^ 

•••   am   ttm+l 

a«2 

a«3 

...  a„. 

•••    ttnn    ttrvn+l 

a,2 


an 

ai3  ••• 

air 

•  ••  a.|,j  ai,,^i 

^21 

a23  ••• 

a2. 

•••  a2„  a2.+i 

«rl 

a,3  - 

a,. 

"•  cim  a„,^i 

am 

a«3  ••• 

a„r 

•••  a„„  a«n+i 

+  --+(-l)"a^^.i 


ttji        (Xj2 

Ctrol  (I9Q 


air 

a2r 


am 

a2n 

a_ 


a^i     a,jj    ...    a^ 
a^i     a„2    ••*    a„y 
in  which  r  has  successively  all  values  from  1  to  n  inclusive. 


=  0, 


98  THEORY  OF  DETERMINANTS, 

84.   We  will   now  select  a  few  examples  to  illustrate  the 
foregoing  processes  from  the  vast  field  of  application. 

I.    To  find  the  condition  that  three  right  lines  shall   pass 
through  the  same  point. 

Let  aiX  +  6i2/  +  Ci  =  0  ■^ 

a2X  +  ^22/  +  ^2  =  0  r  (A) 

G'S^  +  %  +  Cg  =  0  ) 

be  the  equations  of  the  lines  in  cartesian  co-ordinates,  and  let 
a?!,  2/i  be  the  given  point.  Equations  (A)  must  be  satisfied  for 
x  =  Xi,  y  =  yi;  hence 

cti^i  +  hVi  +  Ci 

«2aJi  +  hVi  -f  C2  =  0  ^  •  (JB) 

(^8^  +  ^zjji  4-  C3 


0) 


But  in  that  case,  by  79, 

Iai&2C3l  =  0, 
which  expresses  the  required  condition. 

II.    To  find  the  condition  that  three  points  shall  lie  on  the 
same  right  line. 

Let  (a?i,  2/i)»   ('»2»  2/2)?   (i»8»  ^s) 

be  the  given  points,  and 

ttjx  +  61?/  +  Ci  =  0 

the  equation  of  the  line.     Then 

cfi^^i  +  hVi  +  Ci  =  0, 
«fca;2  +  %2  +  c^  =  0, 
«ba;3 +  6^3  +  03  =  0. 

Whence  the  required  condition  is 


a?i     Vi 

1 

x^    2/2 

1 

^    2/3 

1 

=  0. 


(«) 


APPLICATIONS   AND   SPECIAL   FORMS. 


99 


As  an  application  of  the  present  example,  we  show  that  the 
middle  poirits  of  the  three  diagonals  of  a  complete  quadrilateral 
lie  on  the  same  straight  line. 


The  three  diagonals  being  0(7,  BA^  ByA^,  and  their  middle 
points  F,  D,  E,  we  have  to  show  that  F,  Z),  E  are  on  the  same 
right  line. 

Take  the  vertex  0  as  origin,  and  the  sides  OAi,  OB^  as  axes 
of  reference. 

Put  a^  =  OA,   a^^OA,,   h^  =  OB,  h^^^OB^. 

The  co-ordinates  of  D  are  — ,  -^,  and  the  co-ordinates  of  E 

are   — ,    —  •      The  abscissa  of  F  is  half  the  abscissa  of  (7,  and 

2      2 

the  ordinate  of  F  is  half  the  ordinate  of  C.  Hence  we  have 
to  find  the  co-ordinates  of  0.  The  equations  of  AB^  and  A^  B 
are  respectively 

-+|-  =  1,     or    hcfc ■\- a^y  =  a^hi\ 

«!         02 

— f-  f  =  1 ,     or     hx^-\-aoy=^  a^h^- 
a^     Oi 

Whence  the  co-ordinates  of  G  are 


aib.2 

a. 

W 

a^ 

x  = 

a.A 

as 

? 

y  = 

b, 

&2 

aA 
ai 

h 

«! 

h 

a^ 

W 

aa 

100 


THEORY   OF  DETERMINANTS. 


and  the  co-ordinates  of  F  are 

2  ( 62^12  —  ^i«i )'      2  (^20^2  —  ^lOtl) ' 
Now,  by  equation  {R)  above, 

A  = 


«1 

2 

2 

1 

a, 
2 

&2 

2 

1 

aia2(&2- 

■M 

5i&, 

(02  -  «i) 

1 

=0, 


if  the  three  points  are  on  the  same  straight  line. 


A  = 


1 


4(62«2  — ^l«l) 


a2 


ttj  ttg  (62  —  <^l)        ^1  &2  (<^2  —  <^l)        ^2  %  —  ^1  <^1 


Now  add  the  tliird  column  of  this  determinant  multiplied  by 
—  Qi  to  the  fii'St  column  ;  also  add  the  third  column  multiplied 
by  —  hi  to  the  second  column.     Then 


A  = 


1 


4  (62  0^2  —  ^itti) 


0  0  1 

ag  — «!  h^  —  bi  1 

«!  bi  («!  —  ag)    tti  5i  (6i  —  62)    ^2  «2  —  ^1  «i 


which  is  obviously  zero.  Hence  F,  D,  E  are  on  the  same  right 
line. 

III.  To  ohtaiyi  the  equation  of  a  circle  passing  through  three 
given  points. 

The  general  equation  of  the  circle  is 

{x^  4-2/^)  +2ax  +2by  4-c  =  0. 

If    (a?!,  2/1)?    (a^2,  2/2),   (a^3.  2/3)    are  the  given  points, 

(^1'  +  2/1')  +  2  aa^i  +  2  by,  +  c  =  0, 
(«2'  +  2/2')  +  2  a.r2  +  2  62/2  +  c  =  0, 
W  +  2/3')  +  2aa;3H- 262/3 +  c  =  0. 


APPLICATIONS   AND   SPECIAL   FORMS. 


101 


These  four  equations  are  simultaneous  for  the   parameters 


a,  6,  c'f  hence,  by  79, 


a^  -\-y^  2x  2y  1 

x^'  +  y^'  2x,  22/1  1 

Xz^  +  V'/  ^^2  22/2  1 

^i  +  yi  20^3  22/3  1 


=  0, 


(O 


which  is  the  equation  sought. 

That  equation  C  is  the  required  equation  of  the  circle  deter- 
mined by  («!,  2/i)?  (^2?  2/2)5  (%i  2/3)?  is  obvious  from  the  form  of 
the  first  member.  The  determinant  when  expanded  obviously 
gives  a  function  of  the  second  degree,  and  having  the  charac- 
teristics which  distinguish  the  equation  of  the  circle.  Moreover, 
this  equation  is  satisfied  for  x  =  Xi^  y  —  2/1,  since  in  that  case 
the  determinant  vanishes.  The  same  is  true  if  x^x^^  ^  =  2/2» 
or  a;  =  a;3,  2/  =  2/3- 

IV.  To  find  the  relation  connecting  the  mutual  distances  of 
four  points  on  the  circle. 

We  must  have,  if  the  points  are  (a?i,  2/1) ,  {x^^  y^}-,  {x^-,  2/3) , 
(^4?  2/4)?  ^  determinant  equation  just  like  the  last  one  above, 
except  that  the  first  row  of  the  determinant  will  have  the  sub- 
scripts 1,  the  second  row  the  subscripts  2,  and  so  on,  the  last 
row  having  the  subscripts  4. 

Accordingly,  multiplying  together 


Xi'  +  Vi' 

-2x, 

-22/1     1 

X 

1     x^    2/1     «i^  +  2// 

xi^yi 

-2x, 

-22/2     1 

1     X2    2/2     a;2"  +  2/2^ 

^i  +  yi 

-2x, 

-22/3     1 

1     X,    y,    x^^  +  yi 

x^^y^ 

—  2x^ 

-22/4     1 

1     x^     y^     x^'  +  y,' 

which  are  two  different  forms  of  the  first  member  of  equation 
{(J)  above,  we  obtain  the  required  relation 


0  (12)2  ^13)2  (14)2 

(12)2         Q  ^23)2  (24)2 

(13)2  (23)2        0  (34)2 

(14)2  ^24)2  (34)2         0 


=  0, 


102  THEORY   OF  DETERMINANTS. 

in  which 
{12y  =  {x,-x,y-\-{y,-y2)\    {13y  =  (x,-Xsy  +  (y,-y,y, 

and,  in  general,   {iky  is  the  square  of  the  distance  bet^feen 
the  ith  and  kth.  points. 

Expanding  this  determinant  by  63,  III.,   and   adding  and 
subtracting  4(12)2(13)2(24)2(34)2,  we  obtain 

[(12)2  (34)2  _|.  (13)2  (24)2  _  (14)2  (23)2]2 

-4(12)2(13)2(24)2(34)2  =  0. 

Whence 

|[(12)(34)-(13)(24)-(14)(23)] 

[(12)(34)-(13)(24)+(14)(23)]^ 

Xl[(12)(34)  +  (13)(24)-(14)(23)] 

[(12)  (34)  +  (13)  (24)-f(14)  (23)]J  =  0, 

or  (12)(34)±(13)(24)±(14)(23)  =  0, 

which  expresses  the  condition  sought  in  its  simplest  form. 

V.    To  find  the  condition  that  two  given  straight  lines  in  space 
may  intersect. 

(a)  Let  x  —  a  _  y  —  /3  _  z  —  y  .jx 

tti     ~     bi  Ci   '         ^ 

x—ai  _  y  —  Pi  _  g— yi  (2) 

(3^2  ^2  ^2 

be  the  equations  of  the  lines.     If  these  lines  intersect,   the 

^^^^^^  px-\-gy  +  rz  =  d 

may  be  passed  through  them,  and  we  must  have  for  the  first  line 


pa   +ql3  -{-ry=d\ 
pai  +  Q'&i  +  rci  =  0  J  ' 


(3) 
(4) 


and  for  the  second  line 

pai 

pa^-\-qb2-\-rc2=0  ^  (6) 


jpai  +  gft+ryi  =  (i)  (5)  &- 


\S     ■ 
APPLICATIONS   AND   SPECIAL  FORMS.  103 

From  (3)  and  (5) 

i>(a-ai)  +  g(;S-ft)+r(7-yi)  =  0.         (7) 
(4),  (6),  (7)  being  simultaneous,  the  required  condition  is 


tti  bi  Ci 

0,2  V2  C2 

a  —  tti    p  —  pi    y  —  yi 


=  0. 


(6)  If  the  straight  lines  are  given  by  the  equations 
aix -{- biy -{- CiZ  =  di  1 


a2X-\-  bzy  -^02' 


asX  +  b.sy-\-CsZ  =  ds 
a^x  +  b^y  +  C4 


?3Z  =  (^3    I 


(1) 


(2) 


these  four  equations  are  simultaneous  for  the  point  of  inter- 
section (aj,  y,  2J) ,  and  the  condition  of  intersection  is 

*  ^'  • 

'  ai62C3<^4l  =  0. 

VI.    To  jind  the  equation  of  a  plane  passing  through  three 
given  points  {x^,  2/1?  ^i)^  (^2,  2/25  ^2),  (.^3^  Vs^  ^3)- 
Let  the  plane  be 

aiX-^biy-\-CiZ  =  di.  (1) 

We  must  have 


'aiXs  +  biys-hc^Zs=dj_ 


(^) 


Equations   (D)   and   (1)   being  simultaneous  for  the  para- 
meters Oi,  61,  Ci,  di,  we  have  for  the  equation  sought 


X  y  z  1 

Xi  2/1  ^i  1 

^2  2/2  ^2  1 

iC3  2/3  %  1 


=  0. 


0/1  Vic 


104 


THEOKY   OF   DETERMINANTS. 


VII.  An  interesting  application  of  determinants  is  afforded 
by  the  following  problems. 

(a)  To  extend  a  recurring  series  of  the  rtli  order  without 
knowing  the  scale  of  relation. 

As  is  well  known,  a  series  of  the  form 

UQ  +  Uy^x  +  u^x^-^ h  w^.^a;'*-*-  H h  ^n^""  +  — 

is  a  recurring  series  if  the  relation  of  any  r+1  consecutive  co- 
efficients w„,  ?f„_i,  •••  Un_^  can  be  expressed  by  a  linear  equation 
(the  scale  of  relation).  Under  these  conditions  the  series  is 
called  a  recurring  series  of  the  rth  order.  Every  such  series  is 
accordingly  determined  when  2?*  of  its  consecutive  terms  are 
known.  If  all  the  coefficients,  with  the  exception  of  the  2  rth, 
are  known,  this  last  is  easily  found.  By  the  conditions  of  a 
recurring  series 


^^r+1  -^PlUr        +PlUr-i  +  P^U^-2  H hPr-1^2      +  PAh      =  0 

^%-l  +  Pl^%-2+P2W2r-3  +  i>3W2r-4H hPr-l^r      -^ Pr^r-l^^  0 

U2r      +i)l^2r-l  +  i>2^2r-2  +  P3^2r-3H b Pr-i^K+l+ Pr'^r      =0    , 

Now,  by  79, 


(F) 


U^  W^_l     W^_2     Wr-3 

Ur^l  U^  Ur_i      Ur^2 

U2r-\  U2r-2    %r-3    ^2r-4 

^2r  '^2r— 1    '^2r— 2    ^''2r— 3 


U2        Ui 


Us 


lU 


U,.+i  u. 


=  0, 


whence  Wsr  is  found  by  expanding  the  determinant  and  solving 
the  equation. 

To  find  U2r+i  we  have  only  to  increase  each  subscript  by  unity. 

Applying  the  above  process  to  extend  the  series 


l-\-x-\-bx^-^nx^  + 


APPLICATIONS   AND   SPECIAL  FORMS. 


105 


we  find 

5      1 

1 

13     5      1 

u^    13     5 

13     5 

1 

=  0; 

u,    13     5 

=  0; 

^5       W4       13 

W4     13 

5 

Us     ^/4    13 

Uq     Us     W4 

whence  u^  =  Al,  W5  =  121,  16,5  =  365.  The  series  is  accordingly 
1  _}_  a;  _|- 5ar^  _|- 13a^  +  41  ir*  +  121  a;^  +  365a;^  4- •-. 

(6)  To  find  the  generating  function  for  any  given  recurring 
series. 

Since  a  recurring  series  is  always  the  quotient  of  two  integral 
functions,  of  which  the  divisor  is  of  a  degree  higher  by  1  than 
the  dividend,  we  may  find  the  required  generating  function  by 
indeterminate  coefficients,  as  follows  : 

Assume  the  given  series 


Wo+  iiriX-\-u^-\ |-t^x= 


(T) 


1  +PlX  +  P2^-\ ^PrX' 

(after  both  terms  of  the  fraction  have  been  divided  by  the  first 
term  of  the  denominator) . 

From  the  first  r  of  equations  (F)  of  the  preceding  example 
we  can  determine  the  constants  pi,  pz-'-Pr-  ^e  may  therefore 
find  the  scale  of  relation.  We  have  from  equations  {F)y  after 
obvious  interchanges  of  columns, 


Pr  = 


—  u^ 

Ui    " 

'    U^-2 

Ur-1 

-  w.+l 

W2  •• 

'    y'r-l 

Ur 

-  ?*r+2 

M3  .. 

•    U, 

Ur+1 

• 

I 

: 

* 

—  W2r-1 

Uf    " 

•    W2,_3 

W2r-2 

^0 

Ui" 

•     Wr_2 

^*r-l 

Wi 

U2  " 

•     Wr-1 

U^ 

W2 

1h  " 

•    U^ 

Ur+1 

• 

•        ,  , 

^       * 

* 

W._l 

U,    " 

•    W2r-3 

W2r-2 

Having  determined  the  constants  PxiPi'-'Pn  we  need  only 
clear  equation  {T)  of  fractions;   and  then,  equating  the  co- 


106 


THEORY  OF   DETERMINANTS. 


efficients  of  like  powers  of  x,  obtain  the  usual  linear  equations 
from  which  Oq,  aj,  Og ...  a^_i  are  found. 

For  an  example,  let  us  find  the  generating  function  of  the 
series  we  extended  in  the  last  example. 

Put 

Here 

Wo=l,    Wi=l,    U2  =  5,     '". 

Substituting  in  the  second  member  of  (  Ti),  clearing  of  fractions, 
and  finding  the  values  of  ao  and  Oj,  we  find 


f(^)^ 


1 


1— 2cc-3a^ 


85.  The  coefficients  of  the  quotient  Q  of  two  polynomials 
Pi  and  P2,  and  the  coefficients  of  the  remainder  i?,  can  always 
be  expressed  as  determinants  in  terms  of  the  coefficients  of 
Pi  and  Pg- 

The  method  employed  in  the  following  example  is  applicable 
in  general. 

Pj  =  Oo^  +  %^^  +  (^2^^  +  %^  +  <^4^  +  % ; 
P2  =  60^  +  ^1  ^^  +  ^2^  +  &3 ; 

B  =  VoX^  -\- TiX  +r2. 


Let 


Now 
hence 


P,Q-\-B  =  Pi; 


(i>) 


O'A-  &2f?l  +  &l52  +  ^l, 

a«=  Z'89'2  +  ^2* 


APPLICATIONS   AND   SPECIAL   FORMS. 


107 


From  the  first  three  of  equations  {p)  we  can  find  Qq,  gi,  ^2?  and 
then  taking  the  first  three  with  each  of  the  others  in  succession, 
we  obtain  Tq,  r^,  i^.     For  example, 


&o'92  = 


h 

0 

ao 

h 

&o 

(h 

h 

h 

^2 

;  Wn= 


h 

0 

0 

tto 

h 

^>o 

0 

«'i 

h 

hr 

bo 

a2 

h 

h 

h 

^3 

Let  the  student  find  the  remaining  coeflScients. 

86.  The  coefficients  of  any  equation  can  be  expressed  in 
terms  of  the  roots  as  the  quotient  of  two  determinants,  as  fol- 
lows. The  method  employed  is  applicable  in  general.  By 
reference  to  examples  6  and  7,  page  37,  it  is  readily  seen  that  if 

f{x)  =  ic^  —  a^T?  +  a2X  —  ttg  =  {x  —  a){x  —  (3)  (x  —  y)^ 

we  have 


0'    / 


=  -  (^-y)  (7-«)  («-|8)  (x-a)  {X-P)  (x-y) . 


Expanding  the  first  member, 


P    y 


+x 


1      1 

a      P 


1      1 
1 

y 

/3^    / 


1 


•a^l  1 


1    1 

+  a^ 

1     1     1 

^    r 

a      /3      y 

^  / 

a^    /5^    y 

{a^~aiX^  -{-a2X  —  as), 


From  this  identity  the  required  expressions  in  determinant  form 
are  at  once  obtained  by  equating  the  coefficients  of  like  powers 
of  X. 

87.    With  the  aid  of  determinants  we  readily  find  the  sum  of 
the  like  powers  of  the  roots  of  any  equation,  as  follows : 


108 


THEORY  OP  DETERMINANTS. 


Let  Si,  S2,  S3...s„  denote  as  usual  the  sum  of  the  first,  sec- 
ond, ...  nth  powers  of  the  roots  of 


Then  from  the  theory  of  equations  we  have 


0. 


(1) 


Pi  +Si 

2^2         +P1S1    +   S2 
^Pa         +P2S1    +  i>i«2  +    «3 


=  0 
=  0 
=  0 

{n-l)Pn-l  +  Pn-2Sl  +  Pn-sS2-\-Pn-iS3-\-  "'  +  5^-1  =  0 

nPn  +Pn-lSl  +  Pn-2S2  +  Pn^3S3-\ ^  PlSn-l+ S,,=  0 


From  equations  (S)  we  obtain  at  once 


{S) 


Pi 

1 

0     . 

..0     0 

2P2 

Pi 

1      . 

-00 

3i>3 

P2 

Pi     ' 

..0     0 

(w-l)Pn-l      Pn-2      Pn-S     '"Pi       1 
nPn  Pn-1      Pn-2     '"   P2      Pi 

If  in  (1)  the  coefficient  of  of  had  been  p^y  we  should,  of  course, 

have  to  write  in  the  value  of  s„  just  obtained,  {^^^^—\    instead 

\Pj 
of  (  —  1)",  and  pQ  instead  of  1,  for  each  element  of  the  minor 

diagonal  of  the  determinant.     If  n  =  3,  and  n=  4,  the  above 

formula  gives 


So=   — 


Pi     1      0 
2i)2    Pi     1 

3i?3      P2      Pi 


,  and  54  = 


respectively. 


Pi  1  0  0 

2p,  p,  1  0 

3i>3  P2  Pi  1 

^Pa  Ps  P2  Pi 


88.  Equations  (S)  can  also  be  employed  to  give  the  value 
of  the  coefficients  in  terms  of  s„  Sg?  ^3...  ^n?  ^7  solving  these 
equations  for  the  coefficients.     We  find 


APPLICATIONS  AND    SPECIAL  FORMS. 


109 


JP. 


_(-iV 


Si 

1 

0 

0  . 

..  0 

0 

S2 

Si 

2 

0  . 

..  0 

0 

Ss 

S2 

Si 

3   . 

..  0 

0 

^n-3       •'n-4 


n-1 

Sl 


If,  as  before,  the  coeflScient  of  a;**  in  equation  (1)  had  been  Pq, 
we  would  write  in  this  value  ofp^,  ( ^^^ )  instead  of  (^^—  ]  • 
If  n  =  3,  and  7i  =  4, 

1 


i>3  =  -^ 


Sl  1 

0 

52   Sl 

1 

S3  S2 

Sl 

;  i?4= 


Sl 

1 

0 

0 

S2 

Sl 

2 

0 

S3 

S2 

Sl 

3 

Si 

S3 

S2 

Sl 

89.    Any  differential  equation  of  the  form 

2/3  -h  X12/2 +^22/1  +Xsy  =  0, 


(1) 


in  which  y,  2/1?  y^i  Vz  denote  a  function  of  x  and  its  successive 
derivatives  respectively,  and  Xi,  Xg,  X3  are  also  functions  of  ic, 
can  be  reduced  to  an  equation  of  the  next  lower  order,  provided 
a  particular  solution  of  (1)  is  known. 
Let  y  =  z  satisfy  equation  (1).     Then 


Put 


z^  +  X^z.,^-X2Z^-\-X^z  =  0. 


u  =  y^-'-y,     v  =  zu. 


(2) 


Then,  as  above,  denoting  derivatives  by  subscripts,  we  have 

—  V  -\-zyi-Ziy  =  0. 

—  Vi-^zy2-Z2y=0. 

—  '^2  +  ^ys  +  Ziy2  —  2=22/1  —  23?/  =  0. 


These  three  equations  and  (1)  are  simultaneous  ;  hence 


110 


THEOBY   OF  DETERMINANTS. 


A=    1  Xi    X,  X,y        =0. 

0  0       z  —V  —z^y 

0  2;      0  —Vi—Z2y 

z  zi    -Z2  —Vz-z^y 

Now  multiply  the  fourth  column  of  A  by  - ,  then  add  to  the 

fourth  column  the  first  multiplied  by  z^,  the  second  multiplied 
by  22,  and  the  third  multiplied  by  ^i,  and  we  have 


=  0; 


or  V2Z  +  Vi{Zi—XiZ)-\-v{z2  +  X2z)  =  0, 

which  is  a  differential  equation  of  the  second  order. 


1 

X, 

X2 

0 

0 

0 

z 

—  V 

0 

z 

0 

-Vi 

z 

«i 

-Z2 

-V2 

Resultants,  or  Eliminants. 

90.  If  we  have  given  a  system  of  n  homogeneous  equations 
containing  n  variables,  or,  what  amounts  to  the  same  thing, 
n  non-homogeneous  equations  containing  n  — 1  variables,  it  is 
always  possible  to  combine  these  equations  in  such  a  way  as  to 
eliminate  the  variables  and  obtain  an  equation  of  relation  be- 
tween the  coefficients  of  the  form 


E  =  0. 


(1) 


i?,  when  expressed  in  a  rational  integral  form,  is  called  the 
Resultant  or  Elmiinant  of  the  system.  In  77  and  79  we 
l)ointed  out  the  fact  that  the  equation  i?  =  0  must  hold  be- 
tween the  coefficients  of  a  system  of  equations  if  they  are 
consistent  with  each  other  (simultaneous).  In  the  examples 
of  84  we  repeatedly  found  the  resultant  of  given  systems  of 
equations.  Among  the  most  important  problems  of  elimination 
is  the  following :  to  find  the  resultant  of  two  given  equations, 
containing  a  single  variable. 


,A 


APPLICATIONS  AND   SPECIAL   FORMS.  Ill 

We  consider  first 

Euler's  Method  of  Elimination. 
91.   I.  Given 

f{x)  =Pq3?  -f-  p^x  4-P2  =  0,  (1) 

and  <^{x)  =  q^x^ -^  q^x  +  q^.  (2) 

If  these  equations  have  a  common  root,  we  must  have 

f{x)  4>{x) 

^— ^  =  (a^x  +  as) ,     ^3^.  =  (61a;  +  62) , 

in  which  %,  ag,  61,  62  are  undetermined,  since  r  is  unknown. 
Then 

{\x  +  62)  {p<iX^-\-PiX  +P2)  =  (^la;  +  tta)  (go^+  gi^;  +  ga)- 

Whence  the  equations 

\Pq+    0   -aigo+    0    =0. 
^iPi  +  2>2i>o  -  tti^i  -  a2g'o  =  0. 

&li>2  +  &2i3l  —  «ig2  —  «2gi  =  0. 

0    +&2i>2+    0    —  a2g2  =  0. 


Hence,  by  77,  the  resultant  is 


jR 


=  0. 


Po     0      go     0 

i>i  i>o  gi  go 
P2  Pi  g2  gi 
0    P2    0    g2 

II.    In  general,  let 

f{x)=p,x-  +p^x--'  +p,x^-^  +  ...  +Pr,_^x+p^=0.     (1) 
<^(a;)  =  q,x-  +  gia;'*-'  +  g2a;'*-'  +  -  +  qn-i^  +  g«  =  0.     (2) 

Let  r  be  a  common  root  of  (1)  and  (2),  and  put 

—-^a,af^-^-\.a,x--'+>.>^a^_^x  +  a^=f,{x), 
|^=  6,a--i  +  &2CC-2  +  ...  +\_^x+K^<i>,{x), 


112 


THEORY   OF  DETERMINANTS. 


Id  which  the  coefficients  Oi,  dg?  •••««?  ^i?  &2)  "' K  ^-re  unde- 
termined.    Then 

f,(x)cl>(x)  =  <l>,{x)f{x).  (I.) 

From  the  identity  (I.),  by  the  theory  of  indeterminate  co- 
efficients, we  must  have  m  +  n  homogeneous  equations  between 
the  m  +  n  coefficients  ctj,  ag'-'C*^,  bi,  62  •••6„.  Hence  the  de- 
terminant of  the  system  of  these  m-{-n  equations  must  vanish 
if  (1)  and  (2)  have  a  common  root,  and  the  resultant  sought  is 
accordingly  this  determinant. 

As  an  application  of  Euler's  method,  take  the  following 
example.     To  find  the  conditions  that  must  be  fulfilled  when 

f{x)=poX^+PiX^-hP2X+Ps=0,  (1) 

<f>(x)  =  qoi)(^-\-qiOi^-{-q2X  +  qs=0,  (2) 

have  two  common  roots. 

If  (1)  and  (2)  have  two  common  roots,  two  factors  off{x) 
must  be  the  same  as  two  factors  of  <}){x).     Hence 

(ax+b)  {po^  -^Pi^  +P2^  +P3)  =  (ca;  +  d) {q^  +  qi^-^-q^-^q^) , 
where  a,  b,  c,  d  are  indeterminate  coefficients.     Whence 

apo-\-  0  —cqo+  0  =0. 
«i>i  +  bjio  —  cqi  —  dgo  =  0. 
opa  +  bpi  —  cq.2  —  dqi  =  0. 
«i>3  +  bP2  —  CQs  —  dq2  =  0. 
0  +bps-{-  0  -^^3  =  0. 

From  every  four  of  these  five  homogeneous  equations  we  obtain 
a  determinant  of  the  fourth  order  whose  vanishing  expresses 
one  of  the  required  conditions.  Hence  the  conditions  sought 
are  expressed  by  the  matrical  equation 


Po  Pi  P2  Ps  0 

0  Po  P,  P2  Ps 

Qo  Qi  <?2  ga  0 

0  qo  qi  ^2  Qs 


=  0. 


APPLICATIONS   AND   SPECIAL  FORMS. 


113 


Sylvester's  Dialytic  Method  of  Elimination, 
92.   I.  Given 

PO^  -^Pl^-\-P2^  +  Pi  =  0, 

qoX^  +  qiX  +^2  =0. 


(1) 
(2) 


Multiply  (1)  successively  by  a;^,  ic,  and  (2)  by  a^,  x^^  x.     Then 
we  have  the  following  system  of  equations  : 

PoX^+PiX^+PoX^+PsOi^  =0. 

Pq^  +PiX^  +P2^-\-Ps^  =  0. 

qoO^  +  qj_x^ -}- q^x^  =0. 

qo^:* -\- qiO(^  +  q20!^  =0. 

qo^  +  qiX^-\-q2X  =  0. 

We  may  consider  these  equations  linear  and  homogeneous  with 
respect  to  x^,  «*,  a^,  a^,  x,  considered  as  separate  variables. 
Hence 


E 


PO  Pi  P2  Ps  0 

0  Po  Pi  Pi  Ps 

go  gi  g2  0  0 

0  go  qi  q2  0 

0  0  go  gi  g2 


=  0. 


II.   In  general,  let 

f{x)^poxr-j-p,x^-''-{- 

cf>{x)  =  qoX''  4-gia;"-i  + 


-hqn-1^  +  g»  =  o. 


(1) 

(2) 


If  we  multiply  (1)  successively  by  ic,  x^'"X'^,  and  (2)  succes- 
sively by  ic,  a^-'-ic"*,  we  obtain  a  system  of  m-j-n  equations, 
linear  and  homogeneous,  with  respect  to  a;,  x^,  x^^  ...a;'«+«  con- 
sidered as  separate  variables.  From  these  equations  we  elimi- 
nate the  variables  by  77  and  obtain  the  resultant  in  the  form  of 
a  determinant  of  order  m-\-n. 


114 


THEORY   OF  DETERMINANTS. 


B=       Po  Pi  P2    '"   Pn  Pn+l  Pn+2    '"     =  0. 

0  Po  Pi    '"    Pn-l  Pn  Pn+l 

0  0  Po     '"    Pn-2  Pn-1  Pn 

Qo  qi  92    '"    Qn  0  0 

0  Qo  Ql     •••     Qn-l  qn  0 

0  0  go  •••  qn-2  g„-i  qn 


It  is  evident  from  the  form  of  R  that  the  coefficients  of  (1) 
enter  R  in  the  degree  of  (2),  and  that  the  coefficients  of  (2) 
enter  R  in  the  degree  of  (1). 


Cauchy^s  Modification  of  Bezoufs  Method  of  Elimination. 
93.   I.  Given 

Po^'^+Pl^+P2X+P3=0,  (1) 

and  qoX^  +  qi^^  +  ^2^*^  +  ^s  =  0.  (2) 

Transposing  and  dividing  (1)  by  (2),  we  obtain  successively 

PO  _Pl^+P2^+P8 

go  qi^ -{- q2x -h  qs 

PqX  +i>l  ^P2^-hPn 

qox-\-qi  ~q2^  +  q^ 

Pq^+PiX-^P2  ^  PS^ 

qo^-hqi^  +  q2     qi 

Clearing  these  equations  of  fractions,  we  have 

(Po^i  -  q^Pi)^  +  (i?og2  -  go2>2)«  +  {Poq^  -  qoPs)  =o, 

iPoq2-qoP2)^-^  liPoq-s-qoPs)  +  (2>ig2-giP2)]a;+  {Piqs-qiPs)^^, 
{Poq-s  -  qoPs)^^  +  {Piqs  -  qiPs)^  +  (^2^3  -  ^2^3)  =0. 

Eliminating   o^  and  cc,   regarded  as  distinct  variables,  from 
these  equations  by  79,  we  find 


6- 


APPLICATIONS   AND   SPECIAL  FOKMS. 


115 


R  = 


\P(^qi\  li>o^2l  Ipogsl 

1^0^21    \Poqi\  +  \piq2\    li>i?3l 

IPogsl  li>1^3l  11^2^3! 


=  0. 


The  resultant  is  found  by  this  method  in  the  form  of  an  axisym- 
metric  determinant,*  whose  elements  are  easily  written,  as  we 
shall  show  by  another  example.     Let  the  given  equations  be 


Po^^+Pix 
and                   go^'*  +  5'i^ 

'+P2 
'  +  q2 

x'-\-P3X-hPi=0, 

x^  +  q^x-\-q^  =  0. 

(1) 

(2) 

We  have,  as  before, 

^0 

q^x^ -\- q.;^X' +  q.^a 

q^x  +  qi 
P(^^-\-PiX-^P2 

P2^-\-p^x+p^ 
qzx" -\- q^x -\- q^ 

PsX-\-Pi 
qsx  +  q^ 

' 

-•     (E) 

2JqX^  +  p^a^  +  p^x 
q^a^ -{- q^a^ -\- q^x 

+  ^3 

_Pa 

qi 

Clearing  equations  (E)  of  fractions,  we  have 

\poqi\x'^  +  \Poq2\x^  +  \Poq3\x  +  \poq4\ 


0, 


\poq2\x^  +  UPoQi\-^\PiQ2\^x^  +  UPoqA\  +  [Piq3\lx-{-\piq^\==o, 

ll>0  53laJ'^4-[li>0^4l  +  li>1^3l]a^+[li>ig2H-IP2^3l]^+li>2g4l=0, 

\Poq4\^-{-\piqi\x^-\-\p2qi\x-{-\psq4\  =0. 

Hence,  as  before,  the  resultant  is 


E  = 


\po  qi\ 

\Poq2\ 

\Poqs\ 

\Poq2\ 

\Poq3\  +  \piq2\ 

\poqi\  +  \piqs\ 

IPo  ^sl 

Ipoq^l-hlpiqsl 

\p1qi\-h\P2q3\ 

Ipo  q4\ 

\pi  qi\ 

\P2  qi\ 

\poqi\ 

\pi  qi\ 
\p2qi\ 
IPs  q^l 


=  0. 


For  symmetrical  determinants,  see  107. 


116 


THEORY  OF  DETERMINANTS. 


To  form  this  resultant  directly  from  the  equations,  write  the 
ro  symmetrical  determinants 

\Poqi\    \poq2\    \Poqs\    \poqi\ 
\poq2\    Ipoq^l    \poqi\    Ipiq^l 
\Poq&\    Ipoq^    lihq^l    \p2qi\ 
ipoq^l    Ipiq^l    lp2q^    Ipsqil 

,  and 

\Pi  q2\    \Pi  qsl 
\Pi  q^l    \P2  qs\ 

f 

formed  from  the  coefficients  of  (1)  and  (2)  in  an  obvious  and 
easy  way.  It  is  then  evident  that  B  is  formed  from  these  two 
determinants  by  adding  the  elements  of  the  second  to  the  four 
inner  elements  of  the  first.  If  the  equations  are  of  the  fifth 
degree,  the  student  will  form  the  resultant  in  the  same  way 
from  the  three  determinants 


iPoqil  \Poq2\  \Poq3\  \i>oq^\  Ipoqsl 

iPo^J  IjPogsI  Ipoq^l  Ipoqsl  li^i^sl 

iPo^sl  \Poq^\  IPogsI  IPl^sl  11^2^51 

Ii>og4l  Ipogsl  \piq5\  l^gsl  Ip^qsl 

\Poqo\  Ipiqsl  Ipzqsl  \psq5\  Ip^qsl 


\piq2\  Ipiqsl  \piqi\ 
\piq3\  \piqi\  \p2q4\ 
\piqi\  \p2q4\  Ipzqil 

\p2q3\y 


by  adding  the  third  to  the  middle  element  of  the  second,  and 
then  adding  the  elements  of  the  second  to  the  nine  inner  ele- 
ments of  the  first.     This  process  is,  of  course,  general. 

From  the  preceding  examples  we  see  that  by  Bezout's  method, 
tJie  resultant  of  two  equations,  each  of  the  nth  degree,  is  a  sym- 
metrical determinant  of  the  same  degree  whose  elements  are  either 
determinants  of  the  second  order  or  the  sum  of  such  determinants. 

II.  If  the  two  equations  are  not  of  the  same  degree,  suppose 
we  have  given 

i>oa^*+Pi^^+i>2^+i>3a;+P4  =  0,  (1) 

qox'^-hq^x  +^2  =0.  (2) 

Multiply  (2)  by  x^ ;  the  equations  are  then 

Po^+Pi^^+Pzx'-hPsX-hPi^Oy  (li) 

qox'-\-q,x'-\-q,x'  =0.  (2,) 


APPLICATIONS  AND   SPECIAL  FORMS. 


117 


From  (li)  and  (2i), 

PO^  +Pl  ^  P2^'^  +  P3^  +  j^4^ 

Clearing  these  equations  of  fractions,  we  have 

\po  q2\^  +  \  \pi  ^2'  -  Q'oi>3i  ^  -  (^oi>4  +  qiPs)x  -  qip^  =  0. 

With  these  equations  consider  (2)  multiplied  by  x,  and  (2), 

qQx'^  +  qiX^-^q2X  =  0, 
qooc^-^qix  +  q.^    =0. 

From  these  four  equations  eliminate  x^,  o^,  cc,  and  we  have 


B 


1^0^21 
0 


\poq2\  qoPs        qoPi 

Piq2\-qoP3   qoPi  +  qiPs   qiPi 
qi  -q2  0 


qo 


•qi 


-q' 


0. 


III.    In  general,  let 
f(x)  =poQcr  +piaf-i  +p2a;«-2  4. 


+  Pm~lOC+Pm=0,       (1) 

<f^(xy=  q,x^  +  gio;'-^  +  q,^-^  +  -  +  ^n-i  a;  +  g,  =  0,     (2) 

in  which  m  is  greater  than  n.      Multiply   (2)   by  mf'^ ;  then 
(2)  becomes 

goa;'"  +  gia;«-i  +  g2aj'""^+  •••  +  gH-i^'"*"'*^^  +  gna;"*""-        (2i) 
From  (1)  and  (2i), 

go  qix""-^  +  q2^  ""  +  •••  +  gn-i  a;"*  "-^^  +  g„a;-  «' 

go^j  +  gi     g2aj"-'  -{-  gga^"'  +  •  •  •  +  qn-x^  +  gn«"~"' 


Poa?"~^+;>ia;"-^+  ...  4.p^_2aj_|_p^_^      p^a;'"-"-f-p„+iaj™-"-i+  ...  4-2?^ 


go«'*"''+gi»"''+  -  +gn-2aj+g„-i 


g„a;- 


118  THEORY   OF  DETERMINANTS. 

Clear  these  equations  of  fractions,  and  consider  with  them 
the  following  m  —  n  equations  obtained  from  (2)  bj'  multiply- 
ing it  in  order  by  1,  «,  a^,  •••a;"'""""^, 

qoOf-'^  +  qiixr  '-  +  q2X"'-^-\ hgn-iic"*~"+ ^n^^'""^  =0, 

go«"    +Q'i«"~^H \-qn-i^     +qn  =o. 

From  these  m  equations  the  resultant  is  obtained  by  elimina- 
ting the  m  —  1  successive  powers  of  x  regarded  as  separate 
variables. 

The  Resultant  in  Terms  of  the  Roots, 

94.   Given 

/  =  j9oa;"'-FPia;"-^H f-i?«-ia;+i)«  =  0,  (a) 

<^=go»"+gia;"-'4-  -  +qn-i^-\-qn  =0.  (&) 

If  tti,  a2,  ...a^  are  the  roots  of  (a),  and  /3i,  ^2-,  ...^n  are  the 
roots  of  (6) ,  we  have,  of  course, 

<t>  =  qo{x-(3,){x-p2)'"{^-^n)-  (h) 

Now,    if    in    go«''*  + Q'i^'*~^H hO'n-i^  +  Q'n    we   substitute 

successively  ai,  og,  ...  a^,  </>  takes  the  m  corresponding  values, 
<^(ai),  </)(a2)  ...<^(a^).  With  these  m  values  as  roots  we  can 
form  an  equation  of  the  mth  degree  in  <^.  This  equation  may 
be  found  as  follows.     Forming  the  resultant  of 

PoX'^+PiX'^-'^-^ \-Pmi^+p„,        =0,  (1) 

qoX^'+qiX--^  +"'+qn-iX+qn-<l>=-0,  (2) 

bv  92,  we  have 


APPLICATIONS   AND   SPECIAL  FORMS. 


119 


B,= 


PO  Pi  P2"'    Pn             Pn+l         Pn+2 

0  Po  Pi--'    Pn-1         Pn              Pn+l 

0  0  i>o   •••    Pn-2        Pn~l         Pn 

Qo  ^1  Q2  •••  9n-<f>      0           0 

0  go  Qi  •••  Qn-i     qn-4>    0 

0  0  ^0  —  5n-2     g«-i     qn-4> 


=  0. 


This  is  obviously  an  equation  of  the  mth  degree  in  <^,  whose 
roots  are  <^(ai),  <^(a2),  c^Cag),  •••  </>(a,„).  The  absolute  term  T 
of  this  equation  is  the  product  of  its  m  roots  multiplied  by  a 
factor. 

But  from  the  determinant  i?i, 

Again,  since  JKi  becomes  identical  with  {  —  lyR  of  92,  II., 
when  we  have  made  <^  vanish,  we  see  that 

In  just  the  same  way  we  can  show  that 

2"=  (-i)>V(/80/(ft) -/(/S.); 

and  hence,  after  suitable  interchanges  of  lines, 

95.  These  forms  of  the  resultant  R  may  be  obtained  by 
symmetric  functions,  as  follows  : 

/W=i>oa;"  +  Pia;'^-'+i?2a^""'+  •••  +Pm-iX+p,,=:0,     (a) 
<i>{x)  =  q,x-  +  q^x--^  +  q^x^-''  +  ...  +  q,,^.,x  +  q,,  =  0.      (6) 

Then  aj,  ag,  •••a^  being  the  roots  of  (a),  and  ft,  ft,  ."ft 
the  roots  of  (&), 


120  THEORY   OF  DETERMINANTS. 

Now,  if  (a)  and  (6)  have  a  common  root,  the  product 

/(A)/(/SO-/(A)  =  .P 

must  vanish,  since  in  that  case  some  one  of  the  factors  vanishes. 
The  same  statement  applies  to  the  product 

c/>(ai)<^(a2)...(^(aJ=Pi. 

But  /(ft)  =  po  (ft  -  ai)  (ft  -  a^)  . . .  (ft  -  a  J , 

/(ft)  =Po  (ft  -  ai)  (ft  -  a,)  ...  (ft  -  a„) , 

/(ft)  =Po(ft  -  ai)  (ft  -  a^)  ...  (ft-  aj  ; 

also  </)  (ai)  =  qo  (ai  -  ft)  (ai  -  ft)  . .  •  (aj  -  ^„) , 

</>  (02)  =  go  («^  -  ft)  {0.2  -  ft)  —  (a2  -  ft,)  , 

<^(am)  =  go(a.-ft)(a.-ft)-(an.-ft.). 

P  is  accordingly  made  up  of  mn  factors  of  the  form  ft  —  a,. 
We  may  therefore  write 

P=jOo"n(ft-a,), 

where  r  has  all  integral  values  from  1  to  w,  and  s  has  all 
integral  values  from  1  to  m.  P  is  moreover  a  symmetric  func- 
tion of  the  roots  of  <^  (a;)  =0,  and  can  therefore  always  be 
expressed  as  a  rational  integral  function  of  the  coefficients ; 
and  since  it  vanishes  when  /(a;)  =  0  and  cf){x)  =  0  have  a 
common  root,  and  not  otherwise,  when  P  is  expressed  in  terms 
of  the  coefficients,  P  is  the  resultant  of  (a)  and  (6) .  In  the 
same  way 

Pi  =  go'"n(a,-ft)  =  (-l)-«go"n(ft-a.), 

where  s  and  r  have  the  same  values  as  before.  Hence  we  may 
write  the  resultant 

J?  =  (-l)-go'"/(ft)/(ft)-/(ft.)=l>o''<^(ai)c^(a2)...<^(aJ,  (A) 


APPLICATIONS   AND   SPECIAL  FORMS.  121 

for  both  these  expressions  are  rational  integral  functions  of  the 
coefficients  of  f{x)  and  <^(a^),  which  vanish  when  /(a;)=0 
and  <^(a;)  =  0  have  a  common  root,  and  not  otherwise,  and 
-wrhich  become  identical  when  expressed  in  terms  of  the  co- 
efficients.    The  value  of  R  can  accordingly  be  written 


Properties  of  the  Resultant. 

96.  I.  By  reference  to  the  forms  {A) ,  we  observe  that  the 
coefficients  po?  lh'"Pm  of  equation  (a)  enter  the  resultant  in 
the  7ith  degree,  and  the  coefficients  q^^  Q'i-"5'»  of  (6)  enter  the 
resultant  in  the  mth  degree ;  moreover,  we  readily  see  that 
(_l)""*gy'»j>^'*  is  a  term  from  the  first  form  of  the  resultant,  and 
Pq-  q,^  is  a  term  from  the  second  form ;  hence,  given  two  equa- 
tions of  degree  m  and  n  respectively^  the  order  of  the  resultant  R 
in  the  coefficients  is  m-\-n;  the  coefficients  of  the  first  eqiiation 
are  found  in  R  in  the  degree  of  the  second,  and  the  coefficients  of 
the  second  equation  enter  R  in  the  degree  of  the  first. 

II.  If  the  roots  of  (a)  and  (6)  are  multiplied  by  A;,  R  is 
multiplied  by  A;'"".     Since  each  of  the  mn  binomial  factors  of 

is  in  this  case  multiplied  by  A:,  the  truth  of  the  statement  is 
obvious.  This  result  is  frequentlj'  expressed  by  saying  the 
weight  of  the  resultaiit  is  mn.* 

III.  If  the  roots  of  (a)  and  (b)  are  increased  by  7i,  the  resul- 
tant of  the  transformed  equations  is  the  same  as  the  resultant  of 
the  original  equations.  This,  too,  is  obvious,  for  none  of  the 
factors  of  R  is  changed  when  both  roots  are  increased  or 
diminished  by  the  same  number. 

*  By  the  weight  of  any  term  is  meant  the  degree  in  all  the  quantities 
that  enter  it.     The  weight  of  ab^c^  is  6. 


122  THEORY   OF   DETERMINANTS. 

IV.  If  the  roots  of  (a)  and  (b)  are  changed  into  their  recip- 
rocals, the  resultant  Ei  of  the  transformed  equation  is  (  — lj'"'*i?. 

Putting  y  =  —,    (a)  and  (6)  become  respectively 

X 
</>(2/)  =  Qny""  +  Qn-l  2/""'  +  Qn-2  2/""'  +  " '   +  QiP  +  Qo  ==  0.        {b,) 

Whence 


i2x  =  gri>."n(i-i) 


But 

(aia2---a„)= ,    (/Ji/:^2 '••  Pn)  = » 

hence  the  resultant  of  the  transformed  equations  is  identical  with 
the  resultayit  of  the  original  equations^  or  differs  from  it  only  in 
,  sign,  according  as  mn  is  even  or  odd. 

97.  Of  all  the  methods  of  elimination  given,  the  dialytic 
method  is  the  most  direct.  Another  advantage  of  this  method 
is  that  it  may  obviously  be  employed  to  eliminate  one  of  two 
unknowns  from  a  pair  of  equations,  as  in  the  following  example. 

Given 

Po^-{-Pi  ^^y  +P2^y^  +Psf  =  0, 

qo^  +  qixy  ^q.f    +93^  =0. 

/ 

To  eliminate  x  we  form  the  following  equations  : 

Pox!*  -hPi^^^y  ■i-P2^y^  +P3xy^  =  0, 

Po^  -^Pi^y  -hP2xy--hP3f       =0, 

qox*-\-qix^y-i-{q2y'  -\-<i3)x^  =o, 

QoO^    +qiXy  +  q.y--\-q.;^=^0. 


APPLICATIONS   AND   SPECIAL   FORMS. 


123 


Whence 


Po 

PiV 

i>2/ 

Psf 

0 

0 

Po 

PiV 

P2y^ 

Psf 

5o 

qiv 

q2f-\-q3 

0 

0 

0 

Qo 

QiV 

g2y'-\-q3 

0 

0 

0 

go 

qiv 

g2y'  +  qs 

0, 


an  equation  containing  only  y» 

98.  The  same  method  is  also  frequently  applicable  to  the 
elimination  of  w  —  1  unknowns  from  a  set  of  n  equations,  so  as 
to  obtain  a  final  equation  with  but  one  unknown.  It  will  afford 
the  student  a  good  exercise  to  find  from  the  three  equations 

aia^y-\-a2Xz-^as    =0,  (1) 

yz  —  a^x  =  0,  (2) 

a^xy  +aQX  +a7    =0,  (3) 

a  final  equation  in  y,  as  follows  :  First,  eliminate  x  from  (1)  and 
(3),  and  also  from  (1)  and  (2),  obtaining  two  new  equations 
in  y  and  z.     From  these  equations  eliminate  »,  and  obtain 

a^y'^-^a.^a^  0  a^a^ 

—{a^+ae)a2a7    aia^'^+{a^aT^-{-2a^a5aQ)y+a^ai      0 

an  equation  in  y  of  the  sixth  degree.  ~    ' 

99.  A  further  interesting  application  is  found  in  the  follow- 
ing examples,  in  which  three  variables  are  eliminated  from  as 
many  equations.     Given 

a^i +  3^2  +  ^3  =  0,       x^  =  a^       x}—h^       x^  =  c. 

Multiplying  the  first  equation  successively  by 

a?!,  x^-)  Xq^  a;ia;2iC3, 

and  substituting  from  the  last  three,  we  get 

CL  ~\~  x^  X2  ~j~    a?!  x^  ^^  u , 

b+XiX^  -f-   a!2a;3  =  0, 

C  "7~    37  J  ajg  -j~     X2  X^  ^=  '  J , 

cxi  X2  -\-  hxi  x.^  -\-ax.,x.^z=0.  '' 


124 


THEOKY   OF  DETERMINANTS. 


Eliminating  XiX^y  XiX^^  ajg^a? 


a     1 

1 

0 

b     1 

0 

1 

c    0 

1 

1 

0     c 

b 

a 

=  0, 


Had  we  multiplied  the  first  equation  successively  by 

i^       iC2*^3?        *^1*^3?        ^l«^2? 

we  should  find  by  eliminating     XiX2X^,   Xi,   x^^     x^^ 

0. 


0 

1 

1     1 

1 

0 

c     6 

1 

c 

0     a 

1 

b 

a    0 

If  the  original  equations  are 

Xi-\-X2-\-X^=0^       Xi^=a,       X2^=b,       073^  =  c, 

one  form  of  the  resultant  is  obtained  by  multiplying  the  first 
equation  successively  by 

Xi^  ^2?   "^S?         *^2  ^3  »         •^1*^3?         "^i  "^2  ?        "^1  •^2'''3?        3/j*^2  "^S?        2/13/23/3  j 

and  substituting  from  the  last  three.     Then  by  eliminating 

•'^l  )  *^2  )  3/3  J        flJgflJs)        fl72  3?3}        3/1 37^^        X^X^  X^  ^        X^  X^X^  j        37j  SJg  flJg^ 


we  find 


10  0  0  110  0  0 
0  10  10  10  0  0 
0     0     1110     0     0     0 


0 

c 

6 

0 

0 

0 

1 

0 

0 

c 

0 

a 

0 

0 

0 

0 

1 

0 

6 

a 

0 

0 

0 

0 

0 

0 

1 

0 

0 

0 

a 

0 

0 

0 

1 

1 

0 

0 

0 

0 

b 

0 

1 

0 

1 

0 

0 

0 

0 

0 

c 

1 

1 

0 

=  0. 


APPLICATIONS   AND   SPECIAL  FORMS. 


125 


100.    For  a  final  application  of  the  dialytic  method  we  select 
the  following. 

Given  Vc/oic  +  <^^i      +      V6o^  + ^1  +  ^0  =  0, 

to  free  the  equation  from  radicals,  we  ma\'  proceed  as  follows. 
Put  Vao»  +  ai  =  2/i5     V&oa;-f  61  =  2/2. 

Then  we  get  at  once 

2/1+2/2    +c=0, 


2/1^  — Ooa;  — ai  =  0, 
2/2^  — 60  a;  — &i  =  0. 


From  (1)  and  (3), 


1         0         —boX  —  bi 
1     2/1  +  Co  0 

0         1  yi  +  Co 


=  0. 


(1) 

(2) 
(3) 


(4) 


=  0, 


Eliminating  2/1  from  (2)  and  (4) ,  we  have 


1  2cq  Cq  —\x—hi  0 

0  1  2  Co  CQ—h^x  —  hi 

1  0  —  ofo  — %  0 

0      1  0  —aii!— tti 


which  is  the  equation  sought. 
In  general,  given 

PiVfx  +p,Vf(x)  -hPs'^W)  +  -  -^pJVfM  =  B,    , 

in  which  ri,  Vz-'-r^  are  integers,  and  fi{x),  f{x)  --fnix)  are 
rational  integral  functions  of  x,  we  may  rationalize  the  expres- 
sion as  follows.     Put 


/l(^)  =  2//S  f2{^)  =  2/2^"',     -   /n(a^)  =  Vn 


126 


THEORY   OF   DETERMINANTS. 


Then  we  have  a  system  of  n  equations,  from  which,  together 
with 

we  eliminate  the  n  variables  2/1,  Vit  ••.2/»»  ^^^  obtain  a  resulting 
equation  in  x  without  radicals. 

Discriminant  of  an  Equation, 
101.    I.  Given 

f{x)  =p,x-+p,x--''-\-p^x^'-^+  ...  +i>„_ia;4-i>n  =  0,         (1) 
and  the  first  derivatives  of  f{x) ,  or 
f{x)  =  np^--^-\-  {n-l)p,x^-^+  (^_2)p2»--3+...4.p^_,.     (2) 

Then  the  resultant  R  of  f{x)  =  0  and  f\x)  =  0  is  called  the 
discriminant  of  f(x)  =  0,  since,  if  B  vanishes,  f(x)  =  0  and 
/'(if)  =  0  have  a  common  root,  and  hence  f(x)  =  0  has  equal 
roots. 

Forming  the  resultant  of  (1)  and  (2)  by  92,  we  have 

Pn-2        Pn-1  Pn  0         0       0      . 

Pn-S         Pn~2         Pn-l        i^n        0        0      • 
Pn-i         Pn-3         Pn-2     Pn~l    Pn     0      . 

2Pn-2        JPn-1  0  0  0        0. 

^Pn-S       ^Pn-2       Pn-1  0  0        0- 

4i>„-4      ^Pn-i     2p„_2    p„_i  0       0      . 

I 

in  which  the  first  {n  —  1)  rows  are  formed  from  the  coefficients 
of  (1),  and  the  last  n  rows  from  the  coefficients  of  (2). 

Now  multiply  the  first  row  of  B  by  ?i,  and  subtract  it  from 
the  nth  row  ;  the  7ith  row  becomes 


Po 

Pi 

P2 

0 

Po 

Pi 

0 

0 

Po 

npo 

{n 

-^)P1 

(n 

-2)i)2 

0 

npo 

(n 

-l)i>l 

0 

0 

nPo 

0 


j>i    -2i>2 (^-2)i)„_2    -(n-l)i?„_i    -np^     0 


APPLICATIONS    AND    SPECIAL   FORMS.  12T 

Hence  R  is  at  once  reducible  to  a  determinant  of  order  2n  — 2 
multiplied  \)y  Pq]  calling  this  determinant  A,  we  have 

/ 
Now        i2  =i>o'*-V'(aO/'(a2)/'(a3),  -/'(«„) ;    (94  or  95,  A) 

,   ^     f(x)     nx\^yf(x)  f(x) 

and  since    f\x)=--^-^-^  +^-^-^+^-^~L  j^  ...  ^  :^\±, 


/'(ai)       =i>o(ai  — «2)(ai  — as)         •••  («!  —  »„ -i)  (cti  —  a^) 
/'(as).     =i>o(a2  — ai)(a2~a3)         * ' '  («2  —  «n-i)  ("2  —  aj 

f{o-n-\)  =i^o(an-i  — ai)  (a„_i  — as)  •••  (a„_i  — a^.g)  (a„_i  — aj 
/'(a„)       =Po(an  — ai)(an— eta)  ••'  (««— an-2)  (a»  — a„_i) 


{E) 


If  we  multiply  equations  {E)  together,  we  see  that  the  second 
member  of  the  result  will  contain  the  product  of  the  squares  of 
the  differences  of  the  roots  aj,  ag,  ...a„  of  (1).  Employing  the 
usual  notation  for  this  product,  viz.,  ^(ai,  ag,  ag, •••a„),  we  have 

/(ai)/'(a2)  -fM  =  i-iy '"'-'' PoH(a,,  a,,  a,,  -  a  J  ; 

...   A  =  (-l)^^"-^^i>o''^-^r(ai,  02,  a3,  .••  a„). 

II.  The  discriminant  of  an  equation  can  also  be  obtained  as 
follows : 

/(x)  =  0,   (1)  and  f(x)  =  0;  (2) 

being  simultaneous  equations  when  f{x)  =  0  has  equal  roots, 
the  equation 

nf{x)-  xf'(x)  =  0  (3) 

is  also  consistent  with  (1)  and  (2).  Now  (3)  is  an  equation 
of  the  (n— l)th  degree;  and  finding  the  resultant  of  (3)  and 
f'(x)  =  0,  which  is  also  of  the  {n  —  l)th  degree,  we  obtain  the 
discriminant  A  as  a  determinant  of  order  2n  — 2.  For  an 
example,  we  shall  find  the  discriminant  of  the  cubic 

PqX^  +PiX^-\-P2X  +i>3  =  0. 


128 


THEORY   OF   DETERMINANTS. 


We  have  to  find  the  resultant  A  of  the  equations 

p^x^  +  2j92^'  +  3^)3  =  0, 
^PqX^  +  2piX  +    p2  =  0, 

p^       2^92     3i)3       0        =0. 

3jpo     ^Pi     P2       0 
0       3po     2pi      ^2 

By  the  same  process  we  find  the  discriminant  of  the  biquad- 


ratic 
to  be 


P  =  2h^  +  ^Pi^  +  GP2^  +  4^)3^  -\-P4  =  0 


i^o  ^Pi  ^P2  Pa  0        0      =  0. 

0  Pq  32h  ^P2  P3       0 

0  0  po  Spi  3p2  p-s 

Pi  3^2  3^3  p^  0       0 

0  pi  3p2  3^3  i)4       0 

0  0  2h  3^2  3|)3  i)4 


This  is  accordingly  the  same  as  P  —  27J^  =  0,    where 

/  =  P(,Pi  -  4:p^ps  4-  3^2% 

J  =  P0P2P4  +  ^PiP22h  -  PoPi  -  Pi  Pa  -  pi- 

102.  We  may  show  that  »/=  0  is  one  of  the  necessary  con- 
ditions when  the  biquadratic  P  =  0  of  the  preceding  article  has 
three  equal  roots.     Since 

P=i)oa^  +  4_piar^-j-62)2aj'  +  4i)3a^+P4  =  0  (1) 


*  In  many  processes  it  is  found  more  convenient  to  write  a  given  func- 
tion in  the  form  of  this  equation,  i.e., 

+  ^  (n  —  1 )  p„_2 a:2  4-  n;>«- 1  x  +  ;?„, 
Z  ! 

in  which  each  term  is  multiphed  by  tlie  corresponding  coefficient  in  the 

expansion  of  {a:+l)*».     Any  given  polynomial  can,  of  course,  be  at  once 

reduced  to  this  form. 


APPLICATIONS   AND   SPECIAL  FORMS.  129 

has  three  equal  roots,  two  of  these  will  be  roots  of 

K^4-3i?iar^  +  3i>2a;+i)3  =  0,  (2) 

and  one  of  them  is  a  root  of 

Po^  +  2j9i  0^+^92  =  0.  (3) 

From  (2)  and  (3)  this  root  is  also  found  in 

Pix'^  +  2p.2X-^Ps  =  0.  (4) 

Multiplying  (3)  b}*  o^,    (4)  by  2x^  and  adding,  we  obtain 
ay'iPoX^  +  2piX  H-i>a)  +  2x{pixr  +  2p.2X  -fi^s)  =  0.  (5) 

Now  adding  pa^  +  '^Ih^  H~i^4  to  the  first  member  of  (5),  we 
have,  since  P=0, 

a^(PoX^-\-22h^-\-p2)-\-2x(piX^+2p2X+ps)  -j^Paa^H-  '^Ps^-\-p^=  0. 

Hence,  if  (1)  has  three  equal  roots. 


PqX^-\-2piX-{-P2  =  0, 
PiX^-}-2p2X-\-ps  =  0, 
P2ic2+2j93aj+i>4=0. 

or  J=  0. 


Po     Pi     Pi 

Pi      i>2      i>3 
i>2      P&      P4 


=  0, 


The  other  condition  for  three  equal  roots  of  (1)  is  accordingly 
7=0. 

103.  The  resultant  of  a  system  of  n  homogeneous  equations, 
one  of  which  is  of  the  second  degree,  and  the  remaining  n—l 
are  linear,  may  be  obtained  as  follows.     Given 

P=PQaf-\-p^y^--\-p2Z^-^2qoXy-\-2qiXZ-^2q2yz  =  0,     (1) 
Pi=  a^x  -\-b^y  +CiZ  =  0,  (2) 

P2=  a2X  4-  bzy  -\-C2Z  =  0.  (3) 

Differentiating  (1)  with  respect  to  x,  ?/,  z  in  succession,  and 
remembering  Euler's  theorem  on  homogeneous  functions,  we 
obtain 

P=  x(poX-}-qoy-hqiZ)  +  y(qoX-\-piy-i-q2z) 

+  z(qiX-{-  q2y  -i-PiZ)  =  0.  (4) 


130 


THEORY   OF   DETERMINANTS. 


Equations  (2)  and  (3)  and  (4)  are  simultaneous  homoge- 
neous equations  ;  hence,  by  77,  (4)  must  be  expressible  linearly 
in  terms  of  (2)  and  (3),  and 


^1^1  +  192^2  =  0 


(5) 


is  an  equation  identical  with  (4) .     Equating  the  coefficients  of 
(4)  and  (5),  we  have  the  following  system  of  equations: 

Po«  +  go2/  +  ^i2;-^iai- ^2^^2  =  0,  "I 

qiX-{-q2y+P2^—0iCi—02C2  =  O.   ) 

Now,  taking  equations  (2)  and  (3)  with  equations  (E)^  we 
have  a  system  of  five  homogeneous  equations.  Eliminating 
a;,  y^  z,  Oi,  $2-,  the  resultant  of  (1),  (2),  (3)  is 

i2  =    Pq    go    qi    «i    «2 
go    Pi     g2     &i     h 

qi   q-i   P2   ci    C2 

tti     bi     Gi     0      0 
a2    b2    C2     0      0 

In  general,  let  the  system  of  equations  be 

f{x)  =p,x,^    +P2X2     +P3X3    H ^PnXn'  +  ^qiXiX2 

-{-2q2X^Xa-\ \-2q.^.x^_iX^  =0, 

Pi     =aiXi     -hbiX2     -{-CiXs     -{- "• -\-liX,,     =0 

P2       =  aoXi       +b2X2       -{-C2X^       +"'+kXn        =0 


P„  1=  a^.iXi-^b^_iX2-\-Cn-iXs-\ \-l 


0 


(«) 


We  have,  as  before,  if  /x/  denote  the  differential  coefficient 
of  f{x)  with  respect  to  fl;^, 


a?iA'  +  ^2  A'  4-  xja^j  +  -  +  Xnfx,!  =  2/(0.')  =  0. 


(&) 


Since    (a)    and    (&)    constitute   a   system   of    simultaneous 
homogeneous  equations,   (6)  considered  linear  with  respect  to 


APPLICATIONS   AND    SPECIAL   FORMS. 


131 


the    variables,    must   be  expressible    linearly  in    terms  of   the 
n  —  1  linear  equations  of  (a) .     Hence  (6)  is  identical  with 

Oll\  +  e^P,  +  esPs+"'-h  On-lPn-1  =  0.  (c) 

Equating  the   coefficients  of   (5)    and   (c),  we  obtain  the  7i 
homogeneous  equations 

qi^l  +P2^2  +  qn^S-\ \-q2n-l^n  =  Ml +^2^2  +  MsH H  K-lOn-1, 

q^l  +  qn^2  +  P-A^i-\ h  q-ia-S^i  =  CA-^C202+  C3^3  H K  C„_i^„_i, 


g«-li»l  +  ^2«-ia^2  +  ^3H-3^*3H f-Pni»»=  ^A  +  ?2^,  +  ZAH h^H-A-l- 

These  equations,  together  with  the  n—  1  linear  equations  of 
(a),  form  a  system  of  2n—l  equations  between  x^,  x.2,  •••  x^, 
Oil  $2,  •"  ^n  1-     Hence  the  resultant  of  the  given  system  is 


Pi 

qi 

^2 


qi 

P2 

qn 


q2 
qn 

Ps 


qn-l  q2nl  q-An-S 
Qi          5i  Ci 

0,2  h.2  C2 

a«^i  ^„-l  (^n    I 


qn-l  «1  «2 

q2u~l  h  h 

q-Sn-ti  Ci  Co 

Pn  h  k 

Zl  0  0 

/.,  0  0 


«H-1 


Zn-1 

0 
0 


0     0 


0 


Special  Solutions  of  Simultaneous  Quadratics. 

104.  By  the  help  of  a  special  expedient  we  may  often  solve 
a  pair  of  simultaneous  quadratics  much  more  rapidl3'  and  ele- 
gantly with  determinants  than  by  the  ordinary  methods.  The 
following  examples  will  serve  to  exemplify  the  method  em- 
ployed, and  are,  moreover,  such  forms  as  occur  frequently. 

A.   Find  x  and  y  in 

ttiic  +  ^i?/  _  mj  \ 

'~  ^  T,  .  (1) 


^  +  f  = 


132 


THEORY   OF  DETERMINANTS. 


Let  /  be  such  a  factor  that 


(2) 


From  (2) 


/ 


mi   bi 
nil   62 


fD 
A 


/ 


y 


«!  mil 


1  tti    62 1         ^  I  «i    ^2! 

Substituting  in  the  second  equation  of  (1) 

tD 


fDi 

A  ■ 


rA 


B.    Solve  the  equations 


2/  = 


rA 


±Vi>^+A' 


ttiic  +  Wy  —  niixy 
a^x  -\-  622/  =  maic?/ 


(1) 


Divide  these  equations  member  by  member ;  then,  as  before, 
put 

aiX-\-h^y  =  fmi^^ 

a^x-^h^y^fm^]' 


(2) 


/  I    ^1      &2   I 


y  = 


f  I  «i  »%  I 


I  ai     62  1 
From  the  first  equation  of  (1) 

[a,  I  mi  62  I  4-  &i  I  «i  Wi2l]    I  «!  &2  I 


/  = 


£C  = 


mi   I   ?7li    &2  I  I  «1    *^2  I 
I   tti       62  I  I   «1       ^2 


I  tti  m2 1 


y 


I  mi  69  I 


A  shorter  solution  is  obtained  by  dividing  each  equation  of 

(1)  bv  icy,  and  solving  for  -  and  -. 

X  y 


APPLICATIONS   AND   SPECIAL  FORMS. 
C.    Solve  the  equations 


asX^ -h  b2y^  =  rriz ) 


Write  these  equations 

aiX    +&i2/      =  wi 
a2X '  X -{- b2y '  y  =  m. 


:! 


133 


(1) 


(2) 


mi 

61 

«! 

m. 

Then   x  =  - 

ma  622/ 
A 

;     2/  = 

a2X   mg  1 
A 

(" 

«!        6i 

a2a;  622/ 

We  have 

icA     —  Wi  62  2/  =  —  ma  61, 
mia2a;+    A?/     =     aim^, 
ttg  61a?  —  ai  52.7  =—   A. 

Hence 

A      — mi^a       ^2^1 

= 

0. 

mittg         ^        — aimg 

ag^i     — ai52         ^ 

From  which 

A  =  ±  Vc 

11^63^2 +  &l^  0^2  ^2 

-mi^a2&2* 

Again, 

dio;  +  biy    =mi, 
a26ia;  —  ai622/  =  — ^• 

mi 

6. 

ai     mi 

A 

01*2 

ttg^i  —A 

.  (. 

1  = 

ai        61 
cxa&i  —  ai62 

"'  =  "        A 

;      y  = 

Ai 

D.    Solve  the  equations 

QiX -\- biy  =  Ml  I 

«2aJ  H-  ^22/  +  ^2^72^  =m2)' 

These  equations  we  write 

aiX-\-biy        =mi| 


(1) 


(2) 


134 


THEORY  OF  DETERMINANTS. 


\  nil   h\ 


A 

As  before, 

Whence 


02  + Cay   h, 


:!•) 


(A  +  miCg)^/  — aim2+     mia2    =  0, 
—  hiC^y     +  tti  62  —  (X2^i  —  A  =  0. 


A  +  mjCg       mia2  — %^2 
—  61 C2       (Xi^a  —  0^2^!  — ^ 


=  0, 


a  quadratic  from^which  A  is  found. 


I  mi   62  1 


y  = 


I  Oi  mg  I 
A  -h  m  1 C2 


Example  5  above  can  also  be  solved  by  the  method  of  this 
example. 


E.    Solve  the  equations 

aa?  +  hxy  -\-  cy^  =  d  } 
ex^ -^  fxy -]- gy"-  =  h) 

Equations  (1)  may  be  written 

x^ -{- 2  aixy -{-  biy^  =  m 
xP  -\- 2  a2xy  -\- biy^  =  m^ 


;l 


(1) 


(2) 


by  easy  reductions.  We  introduce  the  factor  2  for  convenience 
in  calculation.  A  solution  analogous  to  D  could  be  given. 
Whatever  the  coefficient  of  xy^  it  can,  of  course,  be  at  once 
reduced  to  the  form  2ai.     We  write  equations  (2) 


x{x  +  a^y)  -\r  y  {aiX-\-  h,y)  =  m^ 
x  (x  +  a2y)  +  y  (a.x -\- b. 


hy)  =  mi  I 
),y)  =  m2  j 


(3) 


Then 


X  = 


Ml    aiX-\-b^y 
mo    a^x-^-b^y 


x-\-aiy    mi  I 
X  -\-  a^y    rrio  \ 


APPLICATIONS   AND   SPECIAL  FORMS. 


135 


where 


We  have 


Whence 


x-\-aiy    aiX-{-biy 
x-\-a2y    a2X-\-h2y 


[A+  \aim2\~\x-{-  I  \m2\y  =  0, 
[ma  —  mi]  a;  +  [  I  «!  mg  I  —  A]  2/  =  0. 


A  +  I  aimal  I  ftimgl 

ma  —  mi         I  a-^m^  I  —  A 


0. 


Solving  this  quadratic, 


Now 


A  =  ±  Vl  aimgP—  1  61  ms  I  {m^—m^, 

\mMy 


X  = 


A  +  I  ai  mg 


Substitute  tliis  value  of  x  in  the  first  of  equations  (2),  and 
we  have 

I  mi 62!^/  2a^\mMlf 


(AH- I  aimgl)'^        A+Uim; 


+  \y^  =  ^1, 


a  pure  quadratic,   from  which  the  value  of  y  can  be  found 
at  once. 

105.  To  the  solutions  of  the  last  article  we  add  the  follow- 
ing, in  which  one  equation  is  a  quadratic  and  the  other  is  a 
cubic. 

Find  the  values  of  x  and  y  in 


a?  -{-xy  -\-y^      mi 

x^  —  xy  -\- y'^  ~~  m2 

oc^-\-y^       =  o? 


(1) 


From  the  first  of  equations  (1) 

•  y  =  \mi  I 
.?/  =  Am2  3 


X  {x -\- y) -\- y '  y  =  \mi 
x(x  —  y)-\-y 


(2) 


136 


THEORY   OF   DETERMINANTS. 


A 

m,   y 

A 

a;  +  2/  mi 

X  =  — 

mo  y 
A 

;    2/  =  - 

x  —  ym^ 
A 

.  (.. 

a;-y  y 

We  have 

icA  --  X  (mi  —  wig)  2^  =  0                              | 

Aa;  (mi  -  mg)  -f  [A  -  A.  (mi  -f-  mg)] 

2/  =  0J 

y.) 


(3) 


Whence 


A  A  (?Hi  —  mg) 

A(mi  — ms)     A  — A(mi  +  m2) 


=  0. 


From  this  equation 


^  ~~  2  ^^1  +  ma  ±  Vl077iim2  —  3mi^  —  'dm/\ . 


Now,  since 


A  =  2/, 


we   have   to   find   the  vahie  of    A    in   order   to   complete   the 
sohition. 

From  equations  (2),  and  the  second  of  equations  (1), 


x-\-y 


Am, 


xy  =  -  (wii  -  ma) 


(4) 


From  equations  (4) ,  and  the  first  of  equations  (2) ,  we  get 


A  = 


Vi  (3  mi  m2  -r  m^) 


and  hence 


y  =  ± 


(wi  +  Wo)  ±  VlOmjma—  3mi^—  3m.j 


V  ^  ( 3  m-i  m2  —  mi) 


X  may  be  found  from  the  second  of  equations  (1),  or  from 
the  first  of  equations  (3) . 


APPLICATIONS  AND    SPECIAL  FOKMS. 


137 


Solution  of  the  Cubic, 
106.    The  general  cubic  equation 

PO^  -i-Pl^  +P2^  +i>3  =  0 

is  always  reducible  to  the  form 

x^  +  qiX  +  q2  =  0. 

We  are  therefore  only  concerned  with  the  solution  of  (2), 
The  determinant  equation 

=  0 


is  identical  with 
We  have 


(1) 

(2) 


X 

«i 

as 

a2 

X 

ai 

ai 

a^ 

X 

a^  —  Sa^  €12  x-\-ai^  -{-  a/  =  0. 


A=   £c  +  ai4-a2     %     (ig 
x-\-ai-\-a2     X  .  ai 

^  +  «1  +  «2      ^2      ^ 


(3) 


hence  a;  -{-  «!  +  ag  is  a  factor  of  A. 

Again,  let  a  be  one  of  the  imaginary  cube  roots  of  unity ; 
then  the  other  is  a^.  Substitute  aja,  ag^^  foi"  %  and  a^  re- 
spectively in  A,  obtaining 


2  /v»  n    ^ 


since  a*" 


a2a' 


A  = 


a^  —  3  aittga^^  +  ^i^  +  %^a^  =  ^j 


CC  +  aitt  +  a^a^  CKja  0^2^ 
JC  +  Ofia  +  aoa^  »  a^a. 
a;  rf  «!  a  +  agtt^     a2a^      ^ 


138  THEORY   OF  DETERMINANTS. 

and  hence  A  is  divisible  by  ic  +  aia  +  aga^  By  substituting 
a^a?  and  a^a  for  a^  and  a^  respectively  in  A,  we  obtain  a 
determinant  A",  which  is  shown  equal  to  A  in  the  same  way 
as  before. 

Hence   a^a?  -\- aza -\- x  is  also  a  factor  of  A, 

Accordingly, 

A  =  A:  (a;  +  ai  -f-  <^2)  (a?  +  otja  +  a^a^)  (x  +  aia^  +  aga)  ?     (4) 

where  A:  is  a  numerical  factor.  Comparing  the  term  a^  of  A 
with  the  term  x^  in  the  second  member  of  (4) ,  we  see  that  k=l. 

.*.  x^  —  3ai a2X-\-  cii  +  ag^  =  (x-\-ai-\-  a^  {x-^-a^a  +  a^ a?) 

(ic  +  aja^  +  aaa).  (5) 

From  (5)  we  have  at  once 

a;  =  — «!  — as,       —  aia  — aga^,       —  ^la^  — aga. 

Now  applying  this  result  to  the  solution  of  (2),  we  put 

gi  =  — Saittg^      ^2  =  0^1  +(^2  'i 
whence 


\  2       ^  4.^21  '       ^2\4^27 

Hence,  finally,  the  roots  of  (2)  are 


^-f-#^       ^^-fWlVft'. 


>      2        XT  +  S? 2 +  >      2  +  \T  +  27 2 


APPLICATIONS   AND   SPECIAL  FORMS.  139 


Symmetrical  Determinants. 

107.  When  we  regard  the  square  of  elements  that  make 
up  a  determinant,  it  is  natural  to  inquire  what  special  proper- 
ties, if  any,  the  determinant  possesses  when  we  suppose  the 
elements  not  all  independent ;  in  other  words,  what  special 
forms  arise  when  we  suppose  certain  relationships  to  exist 
between  the  elements,  and  what  are  their  most  important  prop- 
erties. Among  the  special  forms  very  frequently  met  with, 
especially  in  Geometry,  are  the  Symmetrical  determinants.  The 
symmetry  here  referred  to  is  first,  symmetry  with  respect  to 
the  diagonals^  and  second,  symmetry  with  respect  to  the  inter- 
section of  the  diagonals,  i.e.,  the  centre  of  the  square.  Two 
elements,  so  situated  that  the  row  and  column  numbers  of  the 
one  are  the  column  and  row  numbers  of  the  other,  are  called 
conjugate  elements.  Evidently  the  line  joining  two  conjugate 
elements  a„  and  a,^  is  bisected  at  right  angles  by  the  principal 
diagonal.  If  in  a  determinant  a^^  =  dsri  then  the  determinant 
is  axisymmetric,  or  simply  symmetrical.  The  definition  of  a 
symmetrical  determinant  is  extended  so  as  to  mean  symmetry 
with  respect  to  the  secondary  diagonal  also,  so  that  a  deter- 
minant is  symmetrical  if  for  each  element  there  is  an  equal 
element  so  situated  with  respect  to  its  equal  that  the  line 
joining  the  two  is  bisected  at  right  angles  by  one  of  the  diago- 
nals.    The  following  are  symmetrical  determinants  : 


ai     bi     Ci     di 

? 

a,i     ai2 

ai3 

«14 

? 

a. 

&1 

^^ 

d. 

bi     h,Xyd2 

<Xi2       ^13 

«14 

^24 

^2 

62 

C2 

Cl 

Ci   ,.Cjj "  <    ds 

a-13    ai4 

^24 

«34 

% 

h 

h 

bi 

di    ^2    ds     di 

«14       ^24 

«34 

«44 

^4 

as 

^2 

ai 

108.  We  have  already  had  a  number  of  problems  which 
gave  rise  to  symmetrical  determinants.  The  student  may  refer 
to  the  last  determinant  in  example  IV.,  84,  to  the  first  deter- 
minant of  84,  VII.,  to  the  form  of  the  resultant  obtained  by 
Bezout's  method  of  elimination,  93,  (I.),  and  to  the  value  of 


140 


THEORY  OF  DETERMINANTS. 


«7,   102,    for   illustrations   of    how  symmetrical   determinants 
occur  in  practice.     Again,  we  have 


Ittji  CI22  ^33!  — 


^11^21  "h  Cl'12^22  "f~  ^13^28       ^11<^31  "f"  ^12^32  ~f"  <^13^33 


0^21(^11  + Of 22^  12  +  ^23^13  Ct2i  -\-Ct22  ~f"^23 


^21^*^31  ~l~  ^22%2  H~  <^'23^;i3 


%l%l  +  <^32'^12  +  ^33<^i3       <*31^2lH~%2^22  +  <^33^23  ^31  +  <^32^  +  ^^33^ 

which  is  obviously  symmetrical.     It  is  easy  to  show  that  the 
square  of  any  determinant  is  a  symmetrical  determinant.     Let 

then  we  have  to  show  that  b,s  =  ft^-r* 

br,  =  a,ia,i  +  a,2a,2  +  a,.3«,3  H +  «rna.ni 

bgr  =  ot^i  a,.i  +  ag2  ci,.2  H~  ^«3  ^rs  4"  •  *  •  +  <^,„  ofy^ ; 

whence  the  proposition.     An  obvious  coroUar}^  is  that  any  even 
power  of  a  determinant  is  a  symmetrical  determinant. 

109.  It  is  evident  that  conjugate  lines  (a  row  and  a  column 
having  the  same  number)  in  a  sj'mmetrical  determinant  are 
composed  of  the  same  elements  in  the  same  order.  Consider 
now  two  minors  M  and  Mi  of  any  determinant  such  that  the 
rows  and  columns  erased  to  obtain  M  are  the  columns  and 
rows  erased  to  obtain  M^.     Then 

'  M  = 


a^g 

«/A 

(ifi  '" 

,    and   Jfi  = 

%f 

%9 

%9 

%n 

«..•  - 

anf 

O^ng 

... 

... 



a,f 

Clia 

Now,  if  the  determinant  is  symmetrical,  so  that  a„  =  a^,., 
we  have  M=  iV/i,  and,  in  particular,  A^s  =  -4,r ;  or,  in  a  sym- 
metrical determinant,  conjugate  minors  are  equal.  From  this  it 
follows  at  once  that  the  reciprocal  determinant  is  symmetrical. 
Further,  it  is  evident  that  minors  whose  diagonal  lies  in  the 
principal  diagonal  of  a  symmetrical  determinant  (coaxial  minors) 
are  themselves  symmetrical. 


APPLICATIONS   AND   SPECIAL   FORMS.  141 

110.  We  may  show  that  the  product  of  a  symynetric  deter- 
minant by  the  square  of  any  determinant  is  a  symmetric  determi- 
nant^ as  follows  : 

Let  1  ain  I  be  a  symmetrical  determinant,  and  put 

I  aiJ  X  I  ai^l  =  I  Ci„| ,     and     I  a^J  x  ki^l^  =  \biJ. 
Then 

K  =  aaCa  +  f^i2Ck2  +  afs^^s  +  *••  +  ainCkn-  (1) 

In  (1)  substitute  the  values  of  c^,  Cf,2i  •••  C;^,  and  we  have 

+  (Woia^i  +  a.2zaja  +  «23  «ft3  +    •  •  •    +  <^2n  ^^kn)  (^i2 
+ 

+  («niaa  +  an2a;fc2  +  «n3«A3  +    "•    +  ««««&»)«£„, 

=  (^naa  +  Ct2ia,2  +  «3ia;3  -!-•••+  «ni«in)aa 
+  («i2a<i  +  «22af2  +  a32ai3  +  •••  +  «n2am)a*2 

+ 

Since  a^i,  =  a^^,  this  sum  becomes 

O-kl^il  +  aA2C,-2  +   •••    4-  a^nCfn  =  hi' 

Whence   I  biJ  is  symmetrical. 

From  this  and  108  we  see  that  any  power  of  a  symmetrical 
determinant  is  a  symmetrical  determinant. 

111.  Cauchy's  theorem  for  the  expansion  of  a  determinant, 
example  III.,  63,  assumes  a  somewhat  different  form  when 
the  determinant  is  symmetrical.     Thus,  instead  of 

A  =  (loo A'-  ^aioaokAi„, 
we  have,  when  A  is  symmetrical, 

A  =  aoo  A'-  ^a^Q^Aa  -  2  2a,oa;fco4-ft. 
in  which,  as  before,  i  has  all  integral  values  from  1  to  w,  and 


142 


THEORY   OF  DETEEMIKANTS. 


for   ik   we   write   the   different  combinations  of   the   numbers 


1,2, 


taken  two  at  a  time. 


For  example, 


a 

h 

9 

h 

b 

f 

9 

f 

c 

0  a  b  c 

a  0  h  g 

b  h  0  f 

c  9  f  0 


ahc  -  ap  -  bg"  -  cJi"  +  2fg7u 


=  (Tp  J^h-g^-^e-W-'l  abfg 
=  {af+bg-chy-4.abfg. 


2acfk  —  2bcgh 


112.    Consider  the  determinant 


A  =    ai     bi 


d. 


bi  &2  ^2  <^2  62 

Ci  C2  C3  U3  63 

dl  d,  dg  ^4  64 

61  62  63  64  65 


and  suppose  that 


«i+&i4-Ci+<^i  +  ei  =  61+62+^2+5^2+62=  Ci+Ca-f-Ca+cZg -1-63 
=  (^1+^2+^3+^^4+64  =  ei+e2+C3+e4+e5  =  0. 

Then,  first,  A  =  0 ;  since,  if  we  add  tlie  elements  of  the 
other  rows  to  the  corresponding  elements  of  the  first  row,  the 
elements  of  this  row  all  vanish ;  and,  secondly,  we  can  show 
that  all  the  first  minors  of  A  are  equal. 


B, 


bi 

Ci 

d. 

ei 

C2 

C3 

^3 

63 

d. 

d. 

d. 

64 

62 

63 

64 

65 

and    (7o  =  — 


«! 

Ci 

^1 

61 

61 

C2 

d2 

62 

^1 

d. 

^4 

64 

ei 

63 

64 

65 

The  first,  third,  and  fourth  columns  of  Bi  are  identical  with 
the  second,  third,  and  fourth  rows  of  G^.     By  hypothesis  the 


APPLICATIONS   AND   SPECIAL  FORMS. 


143 


elements  of  the  first  row  of  Cs  are  respectively  —  q,  —  Cg,  —  c?3, 
—  63;  whence 


a  = 


A, 


Ci  C3  U3  63 

Oj  C2  Ct'2  62 

di  dg  ^4  64 

61  63  64  65 

as  was  to  be  shown. 

In  general,  if  in  a  symmetrical  determinant  the  sum  of  the 
elements  in  each  row  is  zero^  the  determinant  vanishes^  and  all 
the  first  minors  are  equal. 

Let 

A  =  I  aoo«iia22  •••  a^J ,     with    a„  =  a,^, 

and  ttio  +  a.i  +  ai2  H +  afn  =  0. 

That  A  vanishes  is  obvious.     Again, 


-^ 


«11 

ai2 

... 

«]*-! 

aU 

«1A+1 

«21 

a22 

... 

«2A-1 

.a2A 

<^2A+1 

... 

... 

... 

... 

... 

... 

C^i-U 

«i-12 

... 

«i-lft-l 

«i* 

^^i-lft-fl 

aa 

-ai2 

\ 

ttiA-l 

«a 

«tm 

«i+ii 

«H-12 

^j+l*-l 

«t+l* 

<^i+lft+l 

''nAi+l 


«!„ 


t+ln 


To  the  ith  row  of  Aq^  add  the  remaining  rows ;  the  ^th  row 
becomes 

Then  to  the  Zcth  column  of  A^  add  the  remaining  columns ; 
the  hih.  column  becomes 


Now,  making  the  I'th  row  the  first  row,  and  the  Ajth  column 
the  first  column,  we  have 


144 


THEORY   OF  DETEKMINANTS. 


A.= 

(_!)<« 

«oo 

Ooi 

^02 

«0*-l 

<^0*+l 

"     «0» 

aio 

«u 

ai2 

<*!*-! 

«!*+! 

••      «!« 

a* -10 

«i-ii 

<*i-12 

«i-l*-l 

<*<-l*+l 

••    ai_ 

«i+10 

«t+ii 

«i+12 

<^i+U-l 

Ctt+lA+l 

"   «i+ 

«nO 

«nl 

Ct„2 

«'nA-l 

Cf'nk+l 

-   ««» 

which  pre 

ves  the  theorem. 

— -^1 


113.  If  cc  be  subtracted  from  each  element  of  the  principal 
diagonal  of  a  symmetrical  determinant,  we  have  a  function  of 
X  which,  equated  to  zero,  gives  an  important  equation.  The 
roots  of  this  equation  are  all  real,  which  may  be  proved  as 
follows.     We  have 


/(«')= 

an-x 

aj2 

ai3 

am 

«21 

a22  —  x 

^23 

«2n 

«31 

aga 

^33- 

X 

«3n 

... 

... 

... 

... 

«nl 

«n2 

^nS 

«n«- 

Then 

-x)  = 

an  +  a; 

a,2 

<^13 

... 

«1« 

ttsi 

a^i  +  x 

«23 

... 

a2« 

ttgi 

%2 

^33- 

X 

... 

ttsn 

«n3 


=  0. 


a„„  +  a; 


a«  =  a. 


0) 


(2) 


Multiplying  (1) 

and  (2), 

/W/(-^) 

= 

Pn-0^ 

P12 

Pl3           "• 

Pin 

P21 

P22-^ 

P23           '" 

P2n 

Psi 

P32 

Pm-^  ... 

Psn 

Pnl 

Pn2 

PnS          '•' 

Pnn- 

=  0, 


Pr.  =Pn- 


(3) 


where 


APPLICATIONS   AND   SPECIAL  FORMS. 


145 


Expanding  the  determinant  of  (3)  by  63,  I., 

\p,J-x'^D^_^-^x'^D^_,-x'^D^_,+  ...  +  {-^Y=0.       (4) 

Now  i)„_i,  D^^o^  Dn-s^  '"•>  being  coaxial  minors  of  Ipinl, 
are  all  sums  of  squares  of  minors  of  I  aj^l  ;  for  consider  one 
of  these  minors 


i>n 


Pff      Pf9 
Vgf     Pgg 

Prf      Prg 


'     Pfr 


Prr    P^t=Ptr 


Z)„_2  iiiay  1>G  obtained  by  squaring  the  array 


*/3 


Vn 


in  which  there  are  n  columns  and  n  —  2  rows.  By  58,  1st, 
Z>„_2  must  be  the  sum  of  products  of  pairs  of  determinants 
which  in  this  case  are  equal ;  hence  D„_2  is  the  sum  of  squares 
of  minors  of  I  ai„l  of  order  n  —  2.  Hence  5Z>„_i,  22>„_2, 
2Z>„_3,  •••,  are  all  positive.  The  signs  of  the  terms  of  (4)  are 
therefore  alternately  positive  and  negative,  and,  by  Descartes' 
Rule  of  Signs  (4),  can  have  no  negative  roots.  Accordingly, 
/(a;)  =  0,  or  (1),  cannot  have  a  root  of  the  form  aV— 1,  for 
then  a?  would  be  negative,  which  we  have  shown  is  impossible. 
Nor  can  (4)  have  a  root  of  the  form  /3+aV— 1  ;  for  if  we 
write  au-"iS=au»  a 22  —  1^  =  ci' ^i  etc.,  the  proof  just  given 
is  applicable. 

The  student  will  find  it  interesting  to  apply  the  preceding 
proof  to  the  particular  case  where  f{x)  is  of  the  third  degree, 


i.e. 


/w= 


a-x 

•      «12 

«13 

=  0, 

«12 

a22  +  x 

«23 

«13 

«23 

^33  + a; 

Ctr.  =  «. 

146 


THEOilY   OF  DETERMINANTS. 


actually  multiplying  f{x)  by  /(—«),  and  expanding  the  result 
to  obtain  the  equation  in  a^,  whose  terms  are  alternately  posi- 
tive and  negative. 

114.    Symmetrical  determinants  of  the  form 


/6 


«i 


97     fe     «i     h 
h     97    /e 


d. 

es 

,     and 

Cs 

d. 

h 

Cs 

a, 

h 

fe 

«! 

ao 

ai 

a^ 

ay 

a., 

ag 

a^ 

«3 

^4 

ttg 

a^ 

a. 

an  -I  a. 


^«+i 


=  P(aia2---a2„_2), 


are  called  ortlio symmetric  or  persymmetric.  That  is  to  say, 
when  each  line  perpendicular  to  either  of  the  diagonals  has  all 
its  elements  alike,  the  determinant  is  persymmetric.  Such'  a 
determinant  can  contain  at  most  2n  —  l  distinct  elements. 
Examples  of  the  occurrence  of  orthosym metric  determinants 
in  practice  are  found  in  84,  VII. 

115.  The  most  important  property  of  orthosymmetric  deter- 
minants is  that  the  determinant  remains  unchanged  when  the 
first  terms  of  the  successive  orders  of  differences  of  its  2  n  —  1 
elements  are  substituted  for  the  elements  themselves.  Consider 
the  following  series  of  numbers,  and  form  the  1st,  2d,  3d, 
•  ••  (2n  —  l)th  orders  of  differences  by  subtracting  Oj^^i  from  a^ 


throughout.     Then  adopting 

the  u 

sual  notation,  v, 

tto     ai      tta 

ttg 

«4 

Ct5       '"     Ci2n-2 

Ai     An 

Ai2 

Al3 

Ai4     •••      Ai2„-3 

A2 

A21 

A22 

A23     •••      A2  2„-4 

Ag 

A31 

Ago     •••      A3  2„_5 

A4 

A41     •••     A4  2„-6 

Aat-a 


APPLICATIONS  AND   SPECIAL  FOEMS. 


147 


We  now  show  that 


A  = 


tto     ao 


tto  Qo 


€12       <^S 


a„   a, 


n+l  "'/1+2 


Ct'n-l 

= 

tto 

Ai 

A2 

A3 

••  A, 

«n 

Ai 

As 

A3 

A4     • 

••  A, 

«u+l 

As 

A3 

A4 

A5     • 

••  A, 

... 

... 

... 

... 

...     . 

..     . 

«2n-2 

A«_ 

lA. 

A„+i 

An+2 

..  A. 

n+l 


If  in  A  the  (n-l)th,  (n-'2)th,  ■ 
from  the  nth,  (n  — l)th,  (?i  — 2)th, 
we  get 

A  = 


ao 

Ai 

«! 

An 

«2 

A12 

column  be  subtracted 
•  column  respectively, 


^ln-2 
Aln-1 

Am 


ttn-l     Ai„_i      Ai^        Ai„+i      •••      Ai2«_3 

Repeating  the  operation  successively,  we  obtain 
A  = 


ao 

Ai        A2 

A3        • 

•      K-l 

«i 

All      A21 

A31       • 

•    A,_ii 

a-j 

A12      A22 

A32       • 

•    A._i2 

... 

...        ... 

...      '  . 



a„_ 

1  Ai,,_i  A2„_i 

A3n-1       • 

•    A„.i„_i 

Operating  in  a  similar  manner  upon  the  rows,  we  get 


A  = 


Ai 


Ai 

A2 


A3 
A4 


A4 


A    1    A 
as  was  to  be  shown. 


A„+i  A„+2 


A„_i 

A„ 
A„+i 


Thus 


8   15 
15  26 


26  1  = 
43  ! 


15  26  43   68 
26  43  68  103 


3 

5 

2 

2 

5 

2 

2 

0 

2 

2 

0 

0 

2 

0 

0 

0 

2' 


148 


THEORY  OF  DETERMINANTS. 


for  we  have 


Similarly, 


3     8     15     26  43  68  103 

5      7      11  17  25  35 

2       4  6  8  10 

2  2  2  2. 


7        0-4 

— 

■5 

= 

7 

-7 

3 

0 

=  ( 

0       -4     -5     -3 

-7 

3 

0 

0 

_4     _5     -3       2 

3 

0 

0 

0 

-5     -3       2        10 

0 

0 

0 

0 

dent  may  show  that 

12       4       7 

=: 

1 

1 

1 

0 

2      4       7      12 

1 

1 

0 

1 

4      7      12     19 

1 

0 

1 

-2 

7     12     19     30 

0 

1 

-2 

5 

1       4       9      16 

=  0. 

1 

8 

27 

64 

4       9      16     25 

8 

27 

64 

125 

9      16     25     36 

27 

64      ] 

125 

216 

16     25     36     49 

64 

125     i 

>16 

343 

=  6^ 


Besides  exhibiting  obvious  simplifications,  these  examples 
show  that  wlien  the  elements  of  a  persymmetric  determinant 
of  the  nth  degree  form  an  arithmetical  progression  of  order 
m*  <  n  —  1 ,  the  determinant  vanishes  ;  and  if  the  order  of  the 
progression  is  n—  1,  the  determinant  reduces  to  an  nth  power. 


*The 

series  of  numbers 

1 

8 

27 

64 

125 

216 

form  an 

arithmetical  progression  of  the  third  order,  because  the  terms  of 

the  third  order  of  differences  are  alike. 

Thus 

1 

8 

27 

64 

125 

216 

7 

19 

37 

61 

91 

12 

18 
6 

24 
6 

30 
6. 

APPLICATIONS   AND   SPECIAL  FORMS. 


149 


116.  The  conditions  of  the  last  statement  will  always  be 
fulfilled  if  a^  is  a  rational  integral  function  of  k  of  the  mth 
degree,  whose  highest  term  has  the  coefficient  1.  For  then, 
according  to  the  well-known  theorem,  a^^  aj,  ag,  •••  form  an 
arithmetical  series  of  the  mth  order,  of  which  the  mth.  dif- 
ferences will  be  m!.  If,  then,  ?7i  =  n  — 1,  all  the  elements  of 
the  secondary  diagonal  will  be  (n  — 1)!,  and  all  the  elements 
below  it  will  be  zeros.     Whence  the  determinant  equals 

(_l)2(«-i)[(^  _!)]«. 

If  m  <  n  —  1,  the  determinant  of  course  vanishes.  In  either 
case,  instead  of  ao,  ai,  ag,  •••,  we  may  write 

If,  for  example,  p  is  any  given  number,  and 

-\-m\  _{p-{-k  +  m){p-\-k-\-m  —  l)  •••  {p-\-k-{- 1) 


\       m 


A)  +  mN  /p  -f  m  -f  1 

\    m    )  \       m 

A)-hm  +  l\  /p  +  m-f2\ 

\       m       J  \       m       J 


'p-\-2m 


m 


^p  +  2m\        A)  + 


2m  +  l 
m 


et.'") 


117.    Consider  the  determinant 

A 


k 

kr 

ki""       . 

..     A;r"-^ 

kr 

kr' 

ki^      . 

..     kr- 

kr' 

ki^ 

kr""      . 

..     kr--^' 

^y.n-1    ^.^n        J^^+l     ...        ^^n^ 


150 


THEORY   OF  DETERMINANTS. 


whose  elements  are  in  geometrical  progression.  That  A  must 
vanish  is  obvious  at  sight ;  for  dividing  any  column  except  the 
first  by  the  ratio  ?*,  A  is  seen  to  contain  identical  columns. 
Hence  if  the  elements  of  a  persymmetric  determinant  form  a 
geometrical  progression,  the  determinant  vanishes. 

118.   To  the  results  of  the  last  article  we  add  the  following. ' 
Suppose  in 


A  = 


^2 


Cfn-l    Clr 


^n+1 


each  element  divides  every  other  element  whose  subscript  is 
higher  than  its  own,  i.e.,  in  general, 

a^  =  6o  6i  62  •  •  •  6^. 


Then 

A  = 

60 

6061 

606162 

6061 

606162 

bobib^bs 

606162 

60616263 

6061626364 

60616263 

6061626364 

60616.636465 

6o6i62---6^_i     6o6i62---6„     606162  •••6„.f.i     bob^b.^-'-b^^z 

6o6i62---6„_i 
6061   "'b^_ib^ 
bobi   •••6„6„+i 

Now  it  is  obvious  that  60  is  a  factor  of  the  first  row  of  A, 
6061  is  a  factor  of  the  second  row,  606162  is  a  factor  of  the 
third  row,  and  so  on.     Hence 


A=n    br^ 

i=0 


1 
1 

1 

b, 
b, 
bs 

bibo 
bibs 
b,b. 

616263 
626364           . 
636465 

6162  •••6„_i 
6263  •..6„ 
•       b,b,  ...6.^1 

1 

bn 

bnbn+l 

Kbn+lbn+2    • 

•     bX^i-b.^^ 

APPLICATIONS    AND   SPECIAL   FOKMS. 


151 


Skew  Determinants,  and  Skew  Symmetrical 
Determinants. 

119.  We  have  heretofore  shown  (108)  that  the  square  of 
any  determinant  is  a  symmetrical  determinant.  If  we  now 
write  the  determinant  of  even  order 


3 

?>! 

Cl 

ck 

= 

«2 

b. 

Ca 

d. 

«3 

h 

C3 

ds 

a^ 

h 

C4 

ch 

h 

ai 

-d. 

Cl 

h 

02 

-d^ 

C2 

h 

«3 

-d. 

C3 

h 

a^ 

-d. 

C4 

we  get,  by  multiplying  these  factors  together, 


0       '  -  (^162)  -  (M2)  -(«A)-(M3)  -(«A)-(M4) 

(« A)  +  (M2)  ;  0  -  {(^ih)  -  {C2d3)    -  {ciM  -  (c^d,) 

(«A)  +  (M3)    '(«2&3)-f  (^2(^3)  0  —  («A)  — (^3^4) 

(aA)  +  {cA)  ■  (a^b^)  +  (02^4)       (a^b^)  +  (c^d^)  0 

\    .  • 

In  this  determinant  each  element  is  equal  to  its  conjugate 

with  opposite,  sign,  and  the  elements  of  the  principal  diagonal 

are   zeros.      Such  determinants  are  called  skew  symmetHcal. 

In  other  words,  if  in  a  determinant  we  have  a,^  =  —  a^i  and 

af^  =  0,    the   determinant  is  skew  symmetrical.      If   an  is  not 

zero,  we  have  a  skew  determinant.     It  may  be  shown  that  the 

square  of  any  determinant  of  even  order  can  be  expressed  as 

a  skew  symmetrical  determinant.     Thus,  since 


A  = 


ttll 

«12 

(Xl3          Oi4 

«21 

^22 

^23          «24 

... 

... 



«n-ll 

««- 

12 

ttn-lS    «n-14     ••• 

«nl 

«n2 

«n3         a«4         ••• 

^n-ln-3    On^in-2    ^n-ln-1    ^^n-ln 
<^nn-3  <^n»i    2  <^nn-l  <^;m 


152 


THEORY   OF   DETERMINANTS. 


^22  — ^21  ^24         — ^23 


^n-12 
Ctn2 


^ln-2         — ^ln-3         ^In         — ^n-1 
<^2»-2  — ^2»i-3         ^2»  — ^n-1 

^n-ln-2  — <^n-ln-3  ^'n-ln  — <^«-ln-l 


we  have,  after  multiplying  these  determinants  together, 

'  A2  = 


nn—l 


0 

mi2 

Wi3        . 

••     mi„ 

msi 

0 

^23        • 

•        W'2n 

mgi 

WI32 

0     . 

•        Wig, 

m« 


m. 


m„ 


m,.  =  —  m. 


For 


m,4  =  tt,ia;(.2  —  ^i2^kl  +  ^!3^ft4  —  <^!4<^>t3  4-  •••  +  Otfn-l<^*n  ~  <^in ^*n-l? 

and  hence  m^i  =  0,     and     mi;^  =  —  '^h* 

120.    The  consideration   of   skew  determinants  reduces    to 
that  of  skew  symmetrical  determinants,  as  we  shall  now  show^ 

I.    By  47, 


A^") 


<^21       ^22 


Ct2n 

Of' JIM. 


a,^  = 


Now,  since  a^j  =  —  a;^;,  the  determinants  Aq^''^  Aq^''"^^,  Ao^'*""^^ 
•  ••,  are  all  skew  sj'mmetrical,  and  A^"^  is  expressed  in  terms 
of  skew  symmetrical  determinants. 

II.    If,  further,  a^  in  A^"^  is  equal  to  a;,  we  have 

It  will  soon  be  shown  that  a  skew  symmetrical  determinant 
of  odd  order  vanishes.  Accordiugl}',  the  terms  of  this  expan- 
sion in  which  the  degree  of  Ao  is  odd  will  vanish.     Thus 


APPLICATIONS   AND    SPECIAL   FOEMS. 


153 


+  x 


+  3^ 


X    —a 

- 

-h    — c 

= 

0 

a      X      —d    —e 

a 

b       d          X        -f 

b 

c       e       f        X 

c 

0   -d  -e 

+ 

0  -h  -c 

+ 

d    0     -f 

h     0     -/ 

e 

f       0 

0     f 

0  1 

-a 

-b 

— c 

0 

-d 

— e 

d 

0 

-/ 

e 

/ 

0 

0  —a 

-c 

+ 

a     0 

—  e 

c     e 

0 

0 

—a 

-b 

a 

0 

-d 

b 

d 

0 

r  0-/ 

+ 

0  -e  + 

0  -d 

+ 

0  -c 

+ 

0  -b 

+ 

0  -a  ~ 

L/   0 

e     0 

d    0 

c    0 

b    0 

a    0    _ 

=  aj'  +  (/'  +  e2  +  cZ2  +  c2  +  62  +  a^)  a^  +  (ct/*-  5e  +  cd)\ 
The  student  may  show  that 

=  (a2  +  62 +  c2  4-^2)2. 


a 

6 

c 

d 

-6 

a 

-c? 

c 

c 

d 

<x 

-b 

—d     — c     6       a 

Writing  another  skew  determinant  A^,  whose  elements  are 
e,  /,  g,  h,  in  the  same  form  as  A  just  written,  we  see  that 
Ai=  (e2-|-/2  +  fy2  _|_  7^2)2^      jf   ^g  multiply  A  and  Ai  together 

by  rows,   we   get  another  skew  determinant  Ao,  of  the  same 
form  as  A  and  Aj ;  the  value  of  Ag  may  accordingly  be  written 


where 


m 
n 


(m2  + 7^2 +02+p2)2^ 

ae  4-  6/-f  eg  +  dh,         o 
—  af-\-  be  —  di  +  c?^,    ^ 


ag  +  67i  +  ce  —  d/, 
a/i  —  6^  +  c/4-de. 


We  have  then 

AAi  =   (a2+  62  +  C2+  ^2)2  (e2  +  /2^  ^2^  ;^o^o  _  ^^^o_^  ^2^  ^2  _^^2)2^ 
or  (a2  +  52  _|_  c2  +  c?2)      (e2+/2^  ^2_|_  7^2^   ^  ^^^2  +  ^2  _|.  ^2  _|_^2)^ 

which  is  Euler's  theorem. 


154 


THEORY   OF  DETERMINANTS. 


121.  Returning  now  to  the  consideration  of  skew  symmet- 
rical determinants,  let  us  take  the  two  minors  M  and  Mi  of 
109,  and  making  a^^  =  —  a^^^  an  =  0,  the  determinant  itself  is 
skew  symmetrical ;  M  becomes  M\  and  M^  becomes  Mi. 
Now  since  every  element  of  Jf'  equals  each  element  of  Mi 
with  contrary  sign,  or  since 


Jf' 


and   Ml 


-a, 
—a. 


where  m,  as  before,  is  the  order  of  the  minors,  i.e.,  the  con- 
jugate minors  of  a  skew  symmetrical  determinant  are  equal  if 
m  is  even  ;  but  if  m  is  odd,  the  conjugate  minors  are  equal,  with 
contrary  signs. 

In  particular,  if  n  is  odd,    Aij,  —  A^^. 

But  if  n  is  even,  Aij,  =  —  A^i. 

122.    If  the  skew  symmetrical  determinant 


A  = 


0 


ai2 
0 


-ai3 

is  multiplied  by  (  —  1)'',  we  obtain 

-A=     0 
ai2 

But  since  the  rows  of  A  are  the  columns  of 

A  =  -A,     or     A  =  0. 


-ai2 

-ai3 

0 

-<*23 

«23 

0 

A, 


It  is  obvious  that,  in  general,  the  effect  of  multiplying  a 
skew  symmetrical  determinant  A  of  order  n  by  (  —  1)"  is  to 
change  the  rows  into  columns.     Hence,  when  n  is  odd. 


A=- A. 


APPLICATIONS  AND  SPECIAL  FORMS. 


155 


Therefore  a  skew  symmetrical  determinant  of  odd  order 
vanishes.  In  a  skew  symmetrical  determinant  An  is,  of  course, 
skew  symmetrical ;  hence 

123.  From  121,  where  n  is  odd,  the  reciprocal  determinant 
is  symmetrical ;  and,  if  n  is  even,  the  reciprocal  determinant  is 
skew  symmetrical. 

124.  I.  Consider  the  following  determinant 


A  = 


0 

^14 


0 

ass 

«24 


-ai3 

-«23 

0 

^34 


-«14 
—  «24 

0 


and  the  reciprocal  determinant 


Now,  by  61, 


0 

—  -^13 

-Au 
0 

-A. 


0 

—  ^24 


-^13 
A2S 

0 

-^34 


0 


=  A 


A  = 


A24 

A^ 

0 


I   ^ 


<^2S 
0 


.-.  ^1/ =j=  a23^A,     or 

and  hence  A  is  a  perfect  square. 

II.   We  shall  now  show  that,  in  general,  a  skew  symmetrical 
determinant  of  even  order  is  a  perfect  square. 


Let 


A  = 


0 

«21 


ai2 
0 


•  «1» 

•  ^2n 

.       0 


*2  n+1 


t-nn+l 


''»H-11    "'M+12 


''n+ln 


<^2n-l  1  ^2«-l  2  • '  *   <^2n-l  n  ^hn-l  n+1 
^2n  1        <*2»  2       •  •  *   <^2n  n        ^2»  n+1 


<*12n-l 
^2  2»-l 


0 

^2/»  2n- 


%2» 
<^2  2n 


^2n-l  2n 

0 


ait= 


156 


THEORY   OF   DETERMINANTS. 


Then,  as  above, 
A 


Aa... ,      A, 


=  A  .  Aa. 


11>  <»2n2n« 


(a) 


^2n  1  <*2n  2« 

Now  since  A  is  skew  symmetrical,  and  n  is  even, 


^aii 


^^«2.1=^ 


«2«2n 

A, 


0 ;     and     A 


^12« 


-A 


«2nl' 


«11»  «2«2/i  ' 


or     A  = 


<hnl 


(6) 

Therefore  A  is  a  perfect  square  if  ^au,a2n2n  is  a  perfect 
square.  In  other  words,  a  skew  symmetrical  determinant  of 
order  2?i  is  a  perfect  square  if  one  of  the  next  lower  even  order  is. 
But  it  is  obvious  that  a  skew  symmetrical  determinant  of  the 
second  order  is  a  perfect  square,  and  we  have  shown  above 
in  I.  that  one  of  the  fourth  order  is  a  perfect  square ;  hence, 
by  what  we  have  just  proved,  a  skew  symmetrical  determinant 
of  the  sixth  order  is  a  perfect  square,  and  so  on.  Hence  the 
theorem  is  true  universally. 

For  a  simple  illustration,  let  us  apply  (6)  to  the  following 
determinant : 

A  = 


=  (vx  —  uy-\-  tzf. 


0 

—  X 

X 

0 

y 

t 

z 

u 

-y 

—z 

= 

—X    —y    —z 

—t 

—  u 

0      —t     —u 

0 

—  V 

0 

t         0       —V 

V 

0     -t 

t       0 

As  another  application,  we  establish  the  followitig  relation  : 

9  {a^—a^y  {a^—a^Y  («3— ^i)^  («2— «4)^  («i— "2)^  (as— «4)^ 

The  first  expression  equals  (see  example  7,  page  37) 


a,' 

a,^ 

tti 

1 

X 

1 

—  3% 

3ai2 

-a,' 

ai 

a/ 

a^ 

1 

1 

—  3^2 

3  a/ 

-a/ 

ai 

a/ 

ttg 

1 

1 

-3a3 

3  as' 

-os^ 

a,' 

a,' 

«4 

1 

1 

—  3a4 

3a,2 

-«/ 

APPLICATIONS  AND    SPECIAL  FORxMS. 


157 


0 

(a^-aiy 
{a,-a,y 
(a,-a,y 

{a^-a^y 

0 
{a.,-a,y 
(a^  -  a^y 

(oi  -  a^y 

0 

{a^-a^y 

{a^-a^y 

{0-2- a^y 

0 

{a^-a^y  {ai-a,y 

{a^  —  a^y^  {a^  —  a^y 

0  {a,-a,y 
{a^  —  a^y  0 


{a^-a^y 
(a2-a^y 
ias-a,y 


-^   («2-«3)^ 


=  [(a2-«3)^  (ai-a4)3+  (a^-aj)\a2-a^y-\-  (a^  -a2y(as-aiyf, 

as  was  to  be  shown. 

125.  The  following  proof  of  the  preceding  theorem  has  some 
advantages  over  the  one  just  given.  Let  A  be  a  skew  symmet- 
rical determinant  of  even  order.  Then  A^^^  vanishes.  Let 
pij,  be  the  complementary  minor  of  an,  in  A^^^,  and  hence  a 
second  minor  of  A.     By  60, 


=  0; 


and  since 


ft-.  =  ft.,      PuPuu=-PJ' 


(1) 


(2) 


Expanding  A  by  Cauchy's  theorem,  63,  III.,  in  terms  of  the 
elements  of   the   first  row   and  first  column,   we  have,   since 


^Oll 


0, 


A  =  —  ISttif  ttaft*  =  ^(^li^ik '^^iil^kk')  substituting  from  (2) ,      (3) 

in  which  i*,  k  have  the  values  2,  3,  •••  2n.     From  (3)  we  have 
at  once 

A  =  [2ai,  Vft]^ 

Here  A  is  expressed  as  the  square  of  a  linear  function  of 
the  elements  of  the  first  row.  This  function  is  rational  if 
Vfti  is  rational.  But  pa  is  a  skew  symmetrical  determinant  of 
order  2?i  —  2.  Hence  a  skew  symmetrical  determinant  of  order 
2n  is  a  perfect  square  if  one  of  order  2w  — 2  is.  But  we 
proved  (124,  I.)  that  a  skew  symmetrical  determinant  of  the 


158 


THEORY   OF   DETERMINANTS. 


fourth  order  is  a  perfect  square ;  hence,  by  what  we  have  just 
proved,  one  of  the  sixth  order  is  a  perfect  square,  and  so  on. 

126.    Since 

that  is,  since  A  is  the  square  of  a  linear  function  of  the  ele- 
ments of  the  first  row,  we  see  that  if  A  is  of  the  fourth  order, 
Va  contains  3  terms ;  then,  if  A  is  of  the  sixth  order,  Va 
contains  5.3  terms,  etc.     In  general,  then,  Va  is  the  sum  of 

(2n-l)  (271-3)  ...  5.  3.  1  terms. 

Every  term  of  Va  is,  moreover,  the  product  of  n  elements  of 
A,  in  which  no  subscript  is  repeated.  For,  taking  the  term 
^14  V^44,  for  instance,  we  see  that  it  consists  of  terms  in  which 
neither  of  the  subscripts  i,  4  is  repeated.  But  VAw  will  contain 
a  term  a23  VTss'  ^^  which,  as  before,  -^/y^  contains  none  of  tbe 
subscripts  1, 2, 3, 4 ;  find  so  on.  Hence  Va  is  the  sum-  of 
terms  of  the  form 

ai2  0^34  ^56  •  *  *  <^2n-12n1 

in  which  no  subscript  is  repeated. 

If  A  is  of  the  fourth  order,  for  example,  we  have 


AW  = 


0  ai2  ai3  a]4 

C*2i       v/  tt23  0^24 

(I3I  €ts2  0  0^34 

«41  ^42  «43  0 


,   and   Va^)  =  {ai^a^  ±  ai^a^i  ±  auO^) • 


diu  =  —  a. 


To  determine  which  sign  is  prefixed  to  each  term,  we  observe 
that  since  the  interchange  of  two  subscripts  of  A  amounts  to 
an  interchange  of  two  rows,  and  also  of  two  columns,  and 
therefore  leaves  A  unchanged,  Va  must  be  a  function  in  which 
the  interchange  of  two  subscripts  either  causes  no  change  or 
simply  a  change  in  sign. 


APPLICATIOiSrS   AND   SPECIAL  FORMS.  159 

If  we  consider  any  term  of  Va^''^  as  ai2«34,  which  the  inter- 
change of  the  subscripts  1,2  transforms  into  a2iasi  =  —  ai2a^, 
it  is  obvious  that  Va^^^  does  change  sign  on  interchanging  two 
subscripts.     We  have  then  the  square  root  equal  to 

^12^34  —  ^13^24  ~l~  ^14^23  5  (2) 

for,  if  the  second  term  of  (2)  were  +,  the  interchange  of  2 
and  3,  while  changing  the  sign  of  the  last  term,  leaves  the 
signs  of  the  first  two  unchanged. 

Since  «,*=  — ot^i,  it  is  alwajs  possible  to  so  interchange  the 
subscripts  that  all  the  terms  shall  be  positive.     Thus 

Va<^>  =  ai2a34  +  ai3a42  +  a^aas. 

127.    In  general,  we  proceed  as  follows : 
A  being  a  skew  symmetrical  determinant  of  the  2  nth  order, 
A  contains  the  term 

( — I)     ^'12<^21<^34<^43^56<^*65  •••    ^2n-l  2»  ^2n  2n-l  =   (<^12<^34<^56  *  *  *  ^2n-12n)    • 

Hence  Va  contains  the  term 

The  positive  square  root  of  A  which  contains  T  as  its  first 
term  is  an  important  function,  possessing  many  properties 
analogous  to  the  properties  of  determinants,  and  is  called  a 
Pfaffian.     The  notation 

P=  [1,  2,  ...  2n],     or     (1,  2,  3,  .••  2w), 

has  been  adopted  for  the  Pfaffian.  From  what  precedes,  we 
see  that  the  terms  of  the  Pfaffian  are  obtained  from  the  prin- 
cipal term  by  permuting  the  subscripts  2,  3,  •••  2n  in  all  pos- 
sible ways,  and  changing  sign  with  every  permutation. 

Since  a,j  =  —  a^^,-,  we  may  so  arrange  the  elements  that  every 
term  of  P  is  positive.     Thus  in  the  case  of  A^^^  above  we  have 

Va(^  =  P  =  ai2  a34  -f  ai3  a^^  +  a^  «23-  (i>) 


160  THEORY   OF  DETERMINANTS. 

128.  If  two  subscripts  are  interchanged,  the  sign  of  P  is 
changed.  Let  a^,/3  be  the  terms  of  P  containing  the  element 
a^g.  Then  the  elements  of  /?  do  not  involve  the  subscripts 
r  and  s.     Interchanging  r  and  s,  let  P  become  P'.     Now 

P^=^P'\ 

since  each  square  is  A,  in  which  two  rows  and  also  two  columns 
have  been  interchanged ; 

,\P  =  ±PK 

But  because  of  the  interchange  in  r  and  s, 

ays(^  becomes  —a^,/3', 

or,  since  the  term  a,„/3  of  P  =  —  a^^f3  of  P',  it  follows  that 
P=2  —  P\  as  was  to  be  shown. 

129.  We  shall  now  prove  a  theorem  by  which  we  may  com- 
pute Pfaffians  of  order  2n  from  those  of  order  2n  —  2.* 

Assuming 

VS=(-1)'(2,  3,  ...,  t-1,  z  +  1,  ...,  271),  (1) 

or,  after  making  i  —  2  cyclical  interchanges, 

VS=a+l,  1  +  2,  ...,  271,  2,  3,  ...,  ^-l),  (2) 

where  /?  has  the  same  meaning  as  in  125,  we  show  that 

VSV/y^  =  ft,;  (3) 

and  then  since 

P  =  ai2  V^22  +  «13  V/?33  H f-  ai2n  V/?2«2«5  (4) 

*  There  is  a  difference  in  the  nomenclature.  We  have  here  considered 
the  order  of  the  Pfaffian  to  be  determined  by  the  number  of  subscripts 
involved.  Some  authors  determine  the  order  of  the  Pfaffian  by  the  order 
of  the  terms  in  the  elements.  Thus  (1,  2,  3,  4),  or  j  \an\,  which  we  have 
designated  as  a  Pfaffian  of  the  fourth  order,  is  said  by  some  writers  to  be 
of  the  second  order. 


APPLICATIONS  AND   SPECIAL  FORMS. 


161 


(1,  2,  3,  ...,  2  w)  =  a,,  (3,  ..-,  2n)  +  a^  (4, ...,  2n,  2)  +  -  |    .5) 
+  ai2«(2,3,  ...,2n-l)  j 

To  show  that  upon  the  assumption  (1)  or  (2)  the  equation 
(3)  results,  we  proceed  as  follows  : 


Since 


PiiPkk  —  Hik  > 


the  terms  of  V/Sii  V/S^^i  must  be  equal  each  to  each  to  the  terms 
of  ^ai  or  equal  with  contrary  sign.     The  product 


(-l)'+^(2,  3 
(2 


,  3,  ...,  i-l,  ^  +  l,  .-.,  2n)  |  .g. 

,  3,  ...,  k-1,  A;  +  l,  •••,  2n)  j  ^  ^ 


becomes,  after  a  certain  number  of  interchanges, 

where  p,  q,  r,  •••,  u,  v  denote  the  series  of  numbers 

2,  3,  ...,  2n, 
exclusive  of  i,  k.     Again, 
/8«=(-l)'-^' 


(7) 


^2  2 


"1-12      ^i-13 


t't+i  2    "i+i ; 


^2A-l 


S+l  Ai-1 


^2k+l 
^3k+l 


i-1  A;+l 
i+1  *+l 


(8) 


becomes,  after  the  same  number  of  interchanges  as  were  em- 
ployed to  change  (6)  to  (7), 


(9) 


a^kp 

«*, 

a*r 

"     a,. 

«« 

%P 

a,. 

a^ 

"        %v 

Si 

a,. 

«.. 

a,. 

"     a^v 

a,i 

... 

... 

... 



... 

a^ 

a.2 

a,r 

••        Clvv 

avi 

162  THEORY   OF  DETERMINANTS. 

Now  the  first  term  of  the  product  of  (7)  is 


which  is  identical  with  the  first  pterin  of  the  determinant  (9). 
Whence  the  truth  of  (2)  is  established,  and  (5)  gives  the 
desired  expansion  of  P.  It  is  to  be  noted  that  the  successive 
terms  of  P  are  written  cyclically.  For  example,  A^*^  being  a 
skew  symmetrical  determinant  of  the  fourth  order, 

AW  =  P^=  (1,2,3,4)2, 
and  (1,2,3,4)  =  a-^a^^  +  a^^a^  +  ai4«23- 

A(6)  =  p2^  (1,2,  3,  ..-6)2, 

(1,  2, ...  6)  =  ai2  (3,  4,  5,  6)  +  a,,  (4,  5,  6,  2)  +  a^  (5,  6,  2,  3) 

+  ai5(6,2,3,4)  +  ai6(2,3,4,o) 

=  0ti2  Cfc34  (Xgg  +  fti2  <^35  (^64  "f"  ^^12  ^36  ^45 
+  «13  «45  <*62  4-  ttjg  a46  (X25  +  ^13  Of 42  ttsg 
+  ai4  0t56«23  +  «14  0t52«36  +  ai4«53«62 
4-  a  15  a62  ^34  +  «15  «63  «42  +  «15  «64  «23 
+  aiett^sa^  +  «16«24«53  +  «16«25«34. 

130.  The  student  must  have  already  noticed  the  analogy 
between  determinants  and  Pfaffians  referred  to  above.  The 
following  notation,  based  upon  this  analogy,  is  interesting. 
Since  the  Pfaffian  involves  just  half  the  elements  of  a  skew 
symmetrical  determinant  like  A  of  124,  II.,  we  write  the 
Pfaffian 

P  =:   \  (li2     ^13     ^14      *••      ^1  2n    1  ^I2n 

^23     0^24     •••      0^2  271-1  ^2  2n 

0^34     ...      ^3  2/1-1  Of3  2n 

(hn-2  2n    1    ^2n-22n 
^2»-12n 

which  is  shortened  to 

1 1  «12«23«34  •  •  •  «2r»  -12«  | ,     Or  tO  ff{a^  2»)  5     Or  tO    1 1  ttj  2n 


APPLICATIONS   AND   SPECIAL   FORMS.  163 

In  particular,  we  have  for  a  Pfaffian  of  the  third  order 

Ui    bi    Ci    =ff{aib2Cs)  =  \\ai    h^    C3I 

We  may  accordingly  write  equation  {p),  at  tlie  end  of  127, 

Vaw=  ||ai4|,  or  rather  A^^^  =  \\a^^f  ; 
and  the  general  equation  would  be 

131.  We  must  here  conclude  the  discussion  of  Pfaffians  with 
the  theorem:  a  bordered*  skew  symmetrical  determinant  is 
the  product  of  two  Pfaffians. 

From  equation  (6),  122,  II., 

^  «2nl  —  ^  '  '^«ll.«2n2n' 

/.  Aa,„i=(l,2,...,27i)(2,3,  ...,2n-2,  2n-l),        (1) 

which  proves  the  theorem  when  the  determinant  is  of  odd  order. 
Let  A^**^  be  a  skew  symmetrical  determinant  of  odd  order.    Aa.. 
is  a  skew  symmetrical  determinant  of  even  order,  and  hence 

VA;;r=(-l)'-i(l,2,  ...,  i-l,i  +  l,  ...,7i) 
=  (i+l,  •••,  n,  1,  2,  .-.,  i-1). 

Now  A^"^  being  zero,  we  have,  by  60, 

.        A%,=  Aa,,A«,,. 

.-.Aa^,  =(1  +  1,  ...,n,l,2,  ...,  ^-l)  (2) 

(A;  +  l,...,n,  1,2,  ...,  ^-1), 

which  proves  that  a  bordered  skew  symmetrical  determinant  of 
even  order  is  the  product  of  two  Pfaffians :  for  any  minor  A^.^ 

*  A  bordered  skew  symmetrical  determinant  is  one  in  which  the  minor 
of  one  of  the  corner  elements  is  skew  symmetrical. 


164 


THEORY  OF  DETERMINANTS. 


of  a  skew  symmetrical  determinant  is  evidently  expressible  as 
a  bordered  skew  symmetrical  determinant. 
If 

a,,  =  -a,,^  we  find  by  (1), 


*«61 


<^12       ^13       ^14       ^15       ^16 
^22       ^23       ^24       ^25       ^26 


32       <^33 


a 


34 


*36 


ttfl 


%3       ^54       <^56       <^56 


=  -(1,2,  3,  4,  5,  6)  (2,  3,  4,  5). 


0,a„  =  -a, 


Again,  if 


a..  =  -  a,, 

aa  =  0 


,  we  find  by  (2) 


^«42 


^21  ^%  ^25  ^24 

ttji  a23  <^^15  <^14 

«31  0-33  0-35  a34 

(X51  a^  a^  a^ 


(5,  1,2,  3)  (3,  4,  5,1), 
=  0,a^.  =  -a,. 


as  the  student  can  readily  verify. 


Circulants. 


132.    The  resultant  of 


f{x)  =  aia^  +  a2^4-«'3  =  0, 


<^W=a^ 

1  = 

0, 

Sylvester's  method  (92)  is 

aj    tta    «3     0      0 

= 

cti 

a2         ttg 

0         Oi        ttg       ttg        0 

as 

(Xj       ttg 

0      0     Oj     a2     (/s 

«2 

Oj       tti 

1      0      0-10 

0      10      0-1 

(1) 

(2) 


if. 


APPLIC  ACTIONS   AND   SPECIAL   FORMS. 


165 


Now  tti,  ttg,  ttg  being  the  three  roots  of  unity,  it  is  evident 
(94)  that 

or,  denoting  one  of  the  imaginary  cube  roots  of  unity  by  a, 
the  other  is  a^,  and  we  may  write 

R=f{\)f{a)f{a') 

=  (tti  +  a2  +  (h)  (%a^  +  «£«  +  %)  («!«  +  o.2a^  +  ag) , 
an  equation  exhibiting  the  factors  of  R. 

133.  R  is  evidently  a  symmetrical  determinant  formed  from 
the  elements  ai,  a^,  a^  in  its  first  row,  in  such  a  way  that  the 
last  element  in  every  row  is  the  first  element  in  every  succeed- 
ing row,  and  the  other  elements  are  written  in  order.  Such  a 
determinant  is  called  a  Circulant.*  The  intimate  connection 
of  the  Circulant  of  the  third  order  with  the  cube  roots  of  unity 
was  shown  in  the  last  article.  We  shall  now  prove  that,  in 
general,  the  circulant  of  the  nth  order, 

0=  C(aia2  •••»„)  = 


ai 

as 

ttg       • 

••      ««-l 

a„ 

«« 

ai 

Cfa    • 

••      an-2 

««-i 

a„_ 

iCtn 

ai   . 

••    a»-3 

a«-2 

... 

... 

...   . 

.,     ... 

... 

a. 

a, 

05     • 

..    a, 

^2 

ag 

ag 

^4     . 

"      «n 

«1 

is  the  product  of  all  factors  of  the  form 

a„a,"-i  +  a„_iaf"-2  +  a„_2a,"-«  H h  Cfga,^  +  aga,  +  ai  =  /(aO, 

in  which  a^  is  one  of  the  nth  roots  of  unity,  and  i  accordingly 
takes  successively  all  the  values  1,  2,  •••  w.  In  symbols,  we 
are  to  show 

C  (ai  a2  ttg  •  •  •  a,„)  =  11  (a„  a^"-^  +  a^-i^-^  -\ f-  ^2  «*  +  ^i) 

=  /(ai)/(<'2)/("a)  -/K)- 
Write  another  determinant  of  the  ?ith  order 


The  Circulant  is  of  frequent  occurrence  in  the  Theory  of  numbers. 


166 


THEORY   OF   DETERMINANTS. 


1  ttj  ai  a^ 
1  a2  ai  ai 
1      ag      ai     ai 


1      a„     ai     ai 


ttg 


n-l 


Multiplying  by  rows. 


CA  = 


/(ai)  /(as) 

ai/(«i)        a2/(a2) 
aiVCtti)        ^if{o.2) 


/(a„-0  /(a„) 

a«-i/(a„_i)  a„f{ai) 

aL/(««-i)  a^/Ca.) 

a::l/(a«-0  a:-V(an) 


arv(«i)  arvca^) 

Factoring  this  product, 

CA=/(aO/(a2)-/(a„)A. 
.-.  O      =/(a0/(a2)-/(a„) 

=  II  (anap^  +  a^_ia"-2  H h  a2ai  +  «i) 

For  an  illustration,     x  0  0  0  y 

y  X  0  0  0 

0  y  X  0  0 

0  0  y  X  0 

0  0  0  y  X 

=  (x  +  ai2/)  (x  +  a22/)  (x  +  a32/)  (a?  4-  a,y)  (x  +  a,y) 


=  {^  +  y)fx-h\ 


V5-1   ,  V10  +  2V5 


+ 


V 


-10 


=  oy'  +  2/•^ 

as  was  evident  from  the  beginning. 


APPLICATIONS   AND    SPECIAL   FORMS. 


167 


134.    The  circulant  of  the  fourth  order 


C  = 


a. 


ai     02     ^3 

(X^       Ctj        M'2       ^3 


CI2      Cl^ 


«! 


can  be  expressed  as  a  circulant  of  the  second  order,  as  follows. 
We  have 

-0  = 


aj 

-02 

«3 

-a. 

= 

«i 

^4 

% 

^2 

as 

-a^ 

«1 

-a. 

«3 

«2 

«! 

a4 

a. 

-«! 

«2 

-a.. 

a2 

ai 

0^4 

«3 

^2 

-«3 

a^ 

-ai 

^4 

as 

as 

ai 

The  first  of  these  determinants  is  obtained  by  interchanging 
the  second  and  third  rows,  and  multiplying  by  (  —  1)^;  the 
second  is  obtained  from  the  first  by  reversing  the  order  of  the 
rows,  and  then  reversing  the  order  of  the  columns. 

Multiplying  them  together, 


ai2_2a2a4+«3^ 

2a^a^-ai-a^        0 

0 

2asai—a^—ai 

ai—2a^a.2+a^        0 

0 

0 

0         ^a^a^—a^—a} 

«2^+«/-2aia3 

0 

0        ai-la^a^-^a^ 

'la-n^—ai—a^ 

Whence  expressing   (7^  as  the  product  of  two  minors,  and 
extracting  the  square  root, 

0  = 


ai  ai  —  a4  a2  +  «3  ^3  —  «2  ^4 

«3  «1  —  «2  «2  +  Cll  Ots  —  «4  «4 

ttgai  —  a2«2  +  «i«3  —  Ct4«4 

aitti  —  a4a2  +  ^gas  —  a2a4 

as  was  to  be  shown. 

The  method  employed  in  this  special  case  is  equalh^  appli- 
cable to  show  that,  in  general,  a  circulant  of  order  2n  can  he 
expressed  as  a  circulant  of  the  nth  order. 

We  add  the  following  proof,  however,  which  is  based  upon 
the  fundamental  property  of  circulants. 


1G8 


THEORY   OF  DETERMINANTS. 


We  have  to  show  that 


G: 


(hn-l  ^2»    ^1 


^2n-l    ^2n 
Oj2n-2    ^2n-l 
•  •  •       ^2«-2 


6. 

62 

6»- 

-1 

&n 

6» 

61 

K. 

-2 

6n-l 

&»- 

,J„ 

K 

-3 

&n-2 

... 

... 

... 

... 

h 

6. 

61 

6. 

h 

63 

6„ 

61 

ttg  a^       Ois     '"     «1  «2 

0^2  %      <^4    •••     ^2»         C^i 

where 

hi  =  ttitti  —  a2«a2  +  <*2«-i<^3  — f-  ...  —  a2a2n 

62  =  astti  —  agag  +  o^^a^      — \.  ...  —  a^a^n 

hi  =  a^ai  —  a^a2  +  «3^3      — +•••— «6«2n 

^k  —  ^2*-l^l  —  <^2ft-2^2  "I"  <*2;t-3^3 ^"   ***  ^2;t  ^2n« 

The  first  determinant 


(1) 


(2) 


Now  for  every  2 nth  root  a  of  unity  there  is  one  —a.     Hence 
(2)  may  be  written 

C  =  if  (&na>-'  +  &.-ia>-4  +  ...  4-  63a/  4-  &2a/  +  h)  •  (3) 

i=i 

If  ±  ttj,    ±  ttg,    ±  ttg,    ±04,     ••,    ±a„, 

are  the  2 nth  roots  of  unity,  it  is  evident  that 

222  2 

ttj  ,   02  ,   as  ,    •••,   a,i  , 

are  the  nth  roots  of  unity.  Hence  the  second  member  of  (3) 
equals  the  second  determinant  of  (1),  which  establishes  the 
theorem. 

For  example, 

0  = 


a     h     c    d 

= 

E     F 

d     a    b     c 

F    E 

c    d     a    h 

b    c    d    a 

APPLICATIONS   AND   SPECIAL  FOBMS. 


169 


in  which 


/.  C={a'  +  c'-2bdy-(2ac-b^-d'y. 


Centro-syininetric  Determinants. 

135.  If  we  suppose  a  determinant  to  be  symmetrical  with 
respect  to  the  centre  of  the  square  (centro-symmetric*),  we 
have,  if  the  determinant  is  of  order  2w, 


A  = 


«!! 

ai2 

«ln-l 

«ln 

K 

&I2             • 

••     ^n-1     hn 

«21 

^22 

«2«-l 

«£« 

&21 

K      • 

••     hn~l     hn 

... 

... 

... 

... 

... 

... 



am 

a«2 

««n-l 

Ctnn 

&nl 

&«2            • 

••     Kn-l    Kn 

Kn 

Kn- 

-1 

bn2 

K, 

Clnn 

«n«-l     • 

•    ««2          ««1 

... 

... 

... 

... 

... 

... 



h^ 

hn-\ 

&22 

hi 

«2n 

a2n-l      • 

•    a22         «21 

&ln 

hn^ 

-1 

&12 

6ii 

ttln 

«ln-l      • 

•     Cli2         «11 

We  will  transform  A  as  follows :  add  the  last  column  to  the 
first,  the  (2n  — l)th  to  the  second,  and  so  on,  finally  adding 
the  (n4-l)th  to  the  nth.  Afterward  subtract  the  first  row 
from  the  last,  the  second  from  the  (2n  — l)th,  and  so  on, 
finally  subtracting  the  nth.  from  the  (n  -f-  l)th.     Then 


A  = 


an 

+  &!» 

dm 

+  ^1 

«21 

+   &2« 

«2h 

+  621 

Ofnl 

-\-Kn 

a«. 

+  Ki 

0 

0 

0 

0 

0 

0 

bn 
hi 


bn 


K 

hn 


«21 
«11 


hn 
K 


*  It  may  be  shown  that  the  product  of  any  two  determinants  of  the  nth 
order  is  expressible  as  a  centro-symmetric  determinant  of  the  2  nth  order. 


170  THEORY  OF  DETERMINANTS. 

Hence 

A  =      «!!  +  ^m        «12  +  &1«-1      •••  <hn-l  +  ^12        ^m  +  &11 

0^21  +  hn        ^22  +  &2n-l      *••  «2n-l  +  &22        C'in   +  K 

ttnl  4-   Kn        Cln2+Kn-1      '•'  ^mi-l  +   &n2         «««  4"  &„1 

Cfnn  —   ^»1        <^nn-l  ~~   ^n2      *  *  *  <^n2  —   ^nn-1      ^nl  —   ^nn 

^2n   —  ^21       <^2w-l    —   ^22     *  *  *  <^22  —  02„_i       (X21  —   02„ 

If  A  is  of  order  2  n  H-  1 ,  we  write 


dll  Qq2 

^21  ^22 

h  h 

K  K-1 


«1«  ^<^1        ^U 

Ct2n  "'2         ^21 

Oil  A*i        «!„ 


v-1 

6ln 

&2«-l 

62. 

... 

... 

6„»-i 

Kn 

?. 

k 

0„2 

0,1 

ai2 

a,! 

By   making  just  the   same   transformations   as  before,  wc 
find 


«]1  +  &1«  «12  4-  &1„-1 

^21  +  &2n  «22  +  &2»-l 

««1  +  &«n  «n2  +  Kn-1 

21,  2k 

^nn  —   ^nl  ^nn-1  —  "n2 

Ogn—  621  a2n-l—  &22 

«ln  —  &11  <hn-i.  —   K 


<^2n  H"  ^21        "^2 

2L  r 


ttsi  —  h^ 


APPLICATIONS   AND    SPECIAL  FORMS. 


171 


Collecting  results,  we  have :  a  centro -symmetric  determinant 
equals  the  product  of  two  determinaiits  each  of  the  nth  order, 
if  the  order  of  the  symmetric  determinant  is  2n  ;  if  the  order 
of  the  symmetric  determinant  is  2  n  +  1 ,  the  factors  are  of  order 
n  and  n-\-l  respectively. 

For  an  illustration  we  expand  the  following  determinant : 


a 

h 

c 

d 

e 

f 

9 

h 

h 

a 

d 

c 

f 

e 

h 

9 

c 

d 

a 

b 

9 

h 

e 

f 

d 

c 

b 

a 

h 

9 

f 

e 

e 

f 

9 

h 

a 

b 

c 

d 

f 

e 

h 

9 

b 

a 

d 

c 

9 

h 

e 

f 

c 

d 

a 

b 

h 

9 

f 

e 

d 

c 

b 

a 

a+h  b  +  g  c+f  d-\-e 

b-\-9  «+^^  ^  +  e  c-j-f 

c+f  d  +  e  a  +  h  b+g 

d-he  c+f  b  +  g  a+h 


a  —  h  b  —  g  c—f  d  —  e 

b  —  g  a  —  h  d  —  e  c—f 

c—f  d  —  e  a  —  h  b  —  g 

d—e  c—f  b—g  a—h 


a+h+d+e     &4-gr+c+/ 
b-\-g+c+f    a  +  ^+d+e 


a-^h—d  —  e     b-\-g  —  c  —f 
b-\-g  —  c—f    a  +  h  —  d—e 


a  —  h-\-d—e     b  —  g-\-c—f 
b  —  g  +  c—f    a  —  h-\-d—e 


a  —  h—d-\-e     b  —  g  —  c+f 
b  —g  —  c-\-f    a  —  h—d-\-e 


Continuants. 
136.    Consider  the  three  simultaneous  equations 

(a)  3  iCi  —    iCg  ~  ^  ) 

(b)  iCi-f-4a;2—    ^3  =  0/' 

(c)  X2-\-5xq  =  0  ) 


172  THEORY   OF  DETERMINANTS. 

From  (a), 


,(s-|)-, 


X,  = 


3^^ 

Xi 


From  (p), 


re,  = 


From  (c), 


X2 


^__1. 


3  + 


4-^« 


oJa 


aji  = 


34- 


4  + 


The  value  of  Xi  is  thus  expressed  as  a  continued  fraction. 
If  we  solve  for  Xi  by  69,  we  find 


1 

-1 

0 

-H 

0 

4 

-1 

0 

1 

5 

3 

-1 

0 

1 

4 

-1 

0 

1 

5 

We  see  then  that  a  continued  fraction  may  be  expressed  as 
the  quotient  of  two  determinants. 

We  shall  now  proceed  to  the  application  of  determinants  to 
continued  fractions  in  general. 


137.  From  the  simultaneous  equations 

'  (1)  aiXi  —  X2      =ai 

(2)  a2Xi  +  a2X2  =  Xs 

(3)  asX2-^a^X2  =  Xi 


(»-i) 
(«) 


APPLICATIONS   AND   SPECIAL  FOEMS.  173 

we  obtain  from  (1) 


X,  = 


Xo 
«! 


Substituting  in  this  the  values  of 

2  3  t 


—  J  7 

Xi  X2 


as  obtained  from  (2),  (3),  •••  (n  —  1),  (n),  we  have 

Xx  =  — i — 
ttj  +  a2 


«2  +  «3 


«n-l  +  a„ 


The  value  of  iCi  is  seen  to  be  expressible  as  a  continued 
fraction.  If  we  stop  at  the  nth  quotient,  and  thus  take  the 
nth  convergent  for  the  value  of  cCi,  then  x^^i  and  all  the  suc- 
ceeding x's  must  be  conceived  to  vanish.  In  that  case  Xi  is 
the  continued  fraction. 


jr=^^      02  ttg  a„_i        a„ 


The  consecutive  convergents  to  F  will  be  denoted  by 
Pi       A  P^ 

p 

The  determinant  expression  for  — ^  is  now  found  b}"  making 

Qn 

a„_i  =  0  in  equations  I.,  and  solving  for  x^  by  69.     We  find 


174 


THEORY  OF   DETERMINANTS. 


X,  = 


aj 

-1 

0 

0 

0 

0 

0 

0 

0 

a^  ■ 

-1 

0 

0 

0 

0 

0 

0 

as 

as 

-1 

0 

0 

0 

0 

0 

0 

a* 

a^ 

-1 

0 

0 

0 

... 

... 

... 

... 

... 

... 

... 

... 

0 

0 

0 

0 

0 

ttn-l 

On-X 

-1 

0 

0 

0 

0 

0 

0 

a« 

^n 

ttl 

-1 

0 

0 

0 

0 

0 

0 

ttg 

tt2 

-1 

0 

0 

0 

0 

0 

0 

ag 

% 

-1 

0 

0 

0 

0 

0 

0 

0 

0 

0 

a«-i 

ttn-l 

-1 

0 

0 

0 

0 

0 

0 

^ 

«n 

which  is  the  determinant  expression  sought,  and  hence  is 

Looking  at  numerator  and  denominator  of  this  convergent, 
we  see  that 

^       tti 

and  thus 


dai 


«! 


dQn 


_  da,  ,        Tp         d(\osQ„) 

138.  A  determinant  having  the  form  of  Q„  in  the  preceding 
article  is  called  a  continuant ;  i.e.,  a  continuant  is  a  determi- 
nant in  which  the  elements  outside  of  the  principal  diagonal 
and  the  two  adjacent  minor  diagonals  are  all  zeros,  and  one  of 
these  minor  diagonals  has  each  of  its  elements  —  1 . 

Since 


ttl 

-1 

0 

0 

..     0 

0 

0 

a2 

a^ 

-1 

0 

..       0 

0 

0 

0 

ttg 

as 

-1 

..     0 

0 

0 

0 

0 

0 

0 

•*        ttn-l 

«n~l 

-1 

0 

0 

0 

0 

..       0 

«n 

On 

APPLICATIONS  AND   SPECIAL  FOEMS. 


175 


«1 

ag 

0 

0     . 

..       0 

0 

0 

-1 

a2 

as 

0 

..       0 

0 

0 

0 

-1 

as 

0-4 

..       0 

0 

0 

0 

0 

0 

0 

...    _1 

ttn-l 

a« 

0 

0 

0 

0 

...       0 

-1 

a« 

it  is  immaterial  on  which  side  of  the  principal  diagonal  we 
write  that  minor  diagonal  whose  elements  are  —  1 .  Also  we 
may  write 


ai 

-1 

0 

0 

.       0 

0 

0 

02 

(h 

-1 

0 

..       0 

0 

0 

0 

ag 

03 

-1 

..     0 

0 

0 

0 

0 

0 

0 

••      a«-i 

«n-l 

" 

0 

0 

0 

0 

..       0 

ttn 

ttn 

«! 

1 

0 

0 

0    . 

.      0 

0 

0 

— ag 

a^ 

1 

0 

0    . 

.      0 

0 

0 

0 

—  ttg 

ttg 

1 

0    . 

.      0 

0 

0 

0 

0 

0 

0 

0    . 

•    —  a«-i 

ttn-l 

1 

0 

0 

0 

0 

0    .. 

.      0 

—  a« 

a„ 

We  shall  employ  the  following  notation : 
\aia2as"'a„J 


Thus 


tti     a2     Og    aj 


(h 

-1 

0 

0 

«2 

0^2 

-1 

0 

0 

as 

«3 

— 

0       0         a4       ^4 


Pn 


Returning  to  ^f »  we  may  now  write 

Qn 


176 


THEOKY   OF  DETERMINANTS. 


aj    "3    "^  -   ««      ) 
Qn  /    a2    og   ...   a„      \ 

or,  the  nth  convergent  to  a  continued  fraction  F  is  expressible  as 
the  quotient  of  two  continuants  multiplied  by  the  first  numerator 
ofF. 

139.    Expanding  P„  in  terms  of  the  elements  of  the  last  row, 
we  find  ,  . 

\(h  as      ...      a^J 


=  a„ai 


as     —1       0 


+  a„ai 


"3 

«3 

-1 

0 

0 

"4 

«4 

— 

0 

0 

0 

0 

0 

0 

0 

0 

as 

-1 

0 

0 

"3 

«3 

-1 

0 

0 

a4 

a. 

— 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

^»-l       "'n-1 


0 

0 

0 

0 

0 

0 

0 

0 

0 

"-n-S        "-n-l 


0       a„_2     a„_2 

=  ana/    "«    "^   -   «-^     )-\-a^aJ    ««    «^   -    ^-^     ^ 
Vag  ag      ...      a^_iy  \a2  a^      •.•      a„_2y 

=  «.A-i  +  a,P,_2. 

Similarly, 

Q^=  f    a,    a,   ...   a„      )  =  aj    "^    "«   -   ^-^  ^ 
\ai  a2      ...      a„J  \ai  ag      ...     a„_i/ 

-honf  "^   "^  '"  """-^    ^ 

=  «n  Qn-1  +  an  Q»-2. 


(^) 


(5) 


APPLICATIONS   AND   SPECIAL  FORMS.  177 

140.  It  is  to  be  observed  that  the  equations  {A)  and  (B), 
besides  establishing  the  law  of  formation  of  the  consecutive 
convergents  to  F,  give  the  expansion  of  a  continuant  of  order 
n  in  terms  of  continuants  of  lower  orders.     Thus,  by  (B), 

/    02    ag    a4     \  /    ag    ttg     \  /^    "^    ^ 

\«1     «2/  ^  ^  ^  V«l    <^2/ 

=  aia2<^3<^4  +  aztt^a^  +  aiaga^  +  010204  +  0204. 

141.  Equation  (B)  is,  in  fact,  a  special  case  of  the  more 
general  theorem 

/     ttg     ttg     04    •••    a„       \    _    /     02     Og    •..    "r       W     a^+2    •••     ttn       \ 
\a^        ..-         Or-l/    V^r+2        •••         a„y 

This  is  easily  proved  by  writing  out  the  continuant  of  the 
first  member  in  full,  and  expanding  by  Laplace's  Theorem 
(55)  in  terms  of  the  minors  formed  from  the  first  r  rows  and 
their  complementaries. 

We  may  also  use  {B)  to  obtain  another  expansion  of  Q„. 
Thus 

f     02      «3    •••    ttn       \   _          f     «3      a4   •••    «n       \                   /     ^4      "s   *••   ««  ^ 
\a^     02        •••         On/    ~  "^  Va2    «3        •••         ««/  "^   \«3 «n/ 

as  the  student  may  easily  verify  by  expanding  the  first  member 
in  terms  of  the  elements  of  the  first  column. 

142.  It  will  afford  the  student  an  excellent  exercise  to  take 
the  quotient 


/      Og      Og     ...     O,^        N 

\a^   02     ...      a, J 


178  THEORY  OF  DETERMINANTS. 

and,  with  the  help  of  (C)  of  the  preceding  article,  transform 
it  into  the  continued  fraction 

Ol        «2  «3  ^n 

143.   57  or  62  established  the  theorem 

dai^dap,      dUi^  da^,      da^,,,  dat^ 
Let  A=q^  =  (  "^      '"  ^»     Y 

and  let  i  =  A;  =  1,      p  =  e  =  n. 

Then  c^^A      ^  f  a,      ...  a„_i     A  =  5-i^ 

da^ida^^      V«2  «3  •••     ««-!/         "'i  ' 

where  P„_i  has  the  meaning  assigned  to  it  from  the  beginning. 
^=,(  ^     '"  <^    \  =  Pn, 


Also  dA 

da^ 


Similarly, 


— —  =a2a3  ..•  a„;  =(_l)«-i; 

dain  da„i      ^       ^ 


and  dA 

da„. 


.    /     0.2     as    •••    ttn       \  f     ^S     "4    •••    ttn-l       \ 

"\a1a2      •••      a^jKa^a^      •••     a„_iy 
or  Q„P«-i  -  PnQn-1  =  (-I)'*aia2a3...a,. 


APPLICATIONS   AND   SPECIAL  FOKMS. 


179 


144.    With   the   help   of   determinants   we   may   now  show 
that 


Oi        a2  0-3 


a*-i 


h+i 


+a*-i  +«*  +a*+i         +«„ 


equals 


JCtti 


ajtti 


O-k+l 


as  follows.     The  first  fraction 


F: 


/    ttg    a4   ...    a„     \ 

tti  Va2    «3         •••         «n/ 
/    02    ttg    ...    a„      \ 


+a« 


This  quotient  is  easily  shown  to  be  equal  to 


ai 

-1 

0 

0 

...     0 

0 

0 

0 

0 

...    0    0 

0 

^2 

-1 

0 

...     0 

0 

0 

0 

0 

...    0    0 

0 

"3 

ag 

-1 

...     0 

0 

0 

0 

0 

...     0     0 

X 

0 

0 

0 

0 

...        a;fc_l 

a*-i 

O-k 

0 

0 

...      0     0 

0 

0 

0 

0 

...      0 

-1 

Ctk 

-1 

0 

...      0     0 

0 

0 

0 

0 

...      0 

0 

O-k+l 

«*+l 

-1 

...     0     0 

0 

0 

0 

0 

...      0 

0 

0 

0 

0 

•••      an     «« 

«! 

-1 

0 

0 

...      0 

0 

0 

0 

0 

...    0    0 

«2 

^2 

-1 

0 

...      0 

0 

0 

0 

0 

..0     0 

0 

ag 

a^ 

-1 

...      0 

0 

0 

0 

0 

..0     0 

X 

0 

0 

0 

0 

...    aj^^i 

«*--l 

a* 

0 

0 

..0     0 

0 

0 

0 

0 

...      0 

-1 

a. 

-1 

0 

..0     0 

0 

0 

0 

0 

...      0 

0 

a^+i 

«*+l 

-1 

..0     0 

0 

0 

0 

0 

...      0 

0 

0 

0 

0 

..    a„    a. 

180  THEORY  OF  DETERMINANTS. 

But  this  quotient  obviously  equals 


a,     -1 

0 

0     . 

.      0 

0 

0 

0 

0     . 

..    0 

0 

0     a^ 

-1 

0    . 

..     0 

0 

0 

0 

0     . 

..     0 

0 

0        03 

% 

-1  . 

..      0 

0 

0 

0 

0     . 

..     0 

0 



... 

...     . 



... 

... 

... 

...     . 



... 

0     0 

0 

0     . 

••   ^O.k-1 

a;a,_i 

Xaj, 

0 

0     . 

..    0 

0 

0     0 

0 

0     . 

"     0 

-1 

«* 

-1 

0     . 

..    0 

0 

0     0 

0 

0     . 

..    0 

0 

a*+i 

«*+l 

-1  . 

..      0 

0 

0     0 

0 

0     . 

..     0 

0 

0 

0 

0    . 

'       Cln 

a« 

«!      —  1 

0 

0    . 

..    0 

0 

0 

0 

0     . 

..    0 

0 

a.2     a^ 

-1 

0     . 

..    0 

0 

0 

0 

0     . 

..    0 

0 

0       ag 

«3 

-1  . 

..    0 

0 

0 

0 

0     . 

..    0 

0 

0     0 

0 

0     . 

••  ^k-l 

a^a*-i 

Xaj, 

0 

0     . 

..    0 

0 

0     0 

0 

0     . 

..    0 

-1 

au 

-I 

0     . 

..    0 

0 

0     0 

0 

0     . 

.     0 

0 

"ft+i 

«*+l 

-1   . 

..     0 

0. 

0     0 

0 

0     . 

.     0 

0 

0 

0 

0     . 

.      a^ 

a» 

This  last  quotient  is 

the  continued  fraction 

F'  =  —     "^ 
as  was  to  be  shown. 


Xaj,_i        Xa^       ttj+i 


+a;a,_i  +«*  +«&+! 


145.  In  a  certain  investigation  it  becomes  necessary  to  show 
that  the  denominators  Di  and  D^  of  the  convergents  to  the 
fractions 


b    b 

b 

and 

b       b 

b 

tti  +a2 

+a„ 

an  +a«-i 

+a^ 

are  equal. 


APPLICATIONS 

AND   SPECIAL  FOKMS. 

We  have 

A  = 

ttj 

-1 

0 

0 

..     0 

0 

0 

> 

b 

ag 

-1 

0 

..     0 

0 

0 

0 

6 

as 

-1 

..     0 

0 

0 

0 

0 

0 

0 

..   b 

a„_i 

-1 

A 

0 

0 

0 

0  ' 

..     0 

b 

«n 

A  = 

«« 

-1 

0 

0 

..     0 

0 

0 

6 

«n-l 

-1 

0 

..     0 

0 

0 

0 

6 

«n^2 

-1 

..     0 

0 

0 

0 

0 

0 

0 

..   b 

a. 

-1 

0 

0 

0 

0 

..    0 

b 

ai 

181 


By  reversing  the  order  of  the  columns  in  Da,  and  also  the 
order  of  the  rows,  and  afterward  making  the  rows  the  columns 
in  order,  the  original  determinant  is  unchanged  either  in  sign 
or  magnitude.  But  by  these  transformations  D^  is  changed 
to  Di.     Whence  Dg  =  A>  as  was  to  be  shown. 


146.   The  quotient 


62  C3I 


Itti  62  C3I 
can  be  expressed  as  a  continued  fraction,  as  follows 


I&2  C3I 

\ai  62  C3I 


1       61      Ci 

0     62    C2 

0       63      Cs 

ai    bi    Ci 

^2      &2      C2 

a^    &3    Ca 

1     0       0 

ttg      ^2      I  ^1^2! 
«3      ^3      I  ^1^3! 


0^2      ^2      1^1^21 
%      ^3      I  ^1^3! 


182 


THEORY   OF   DETERMINANTS. 


63         0        0 

l«2&3l       ^2       I^Cal 

0         h     I&1C3I 

= 

&3 

Ia253l 
0 

0 

-63I61C2I 

0 

-1 

!&lC3l 

0        63     I61C3I 

0 

-1 

&2 

-I&1C2I 

0 
-1 

l^C3l 

h  _ 

53I&1C2I 

6IC3I 


\ 


This  process  is  equally  applicable  to  show  that,  in  general, 
the  quotient  of  two  determinants  —  is  expressible  as  a  con- 
tinned  fraction,  provided  only  that 


d^ 


dA. 


Ai=--^,     or     Ai  =  — ^,     or     Aj  = 
dan  ^^in 

147.    The  continued  fraction 

«!         +«2        +«3 


da„ 


or     A,  = 


dAg 


H-^n 


is  evidently  equal  to 


/      a2     ttg        •••    a^      \ 
Vai     02  •••       ct»/ 


For 


a,  -1  0  0  ...  0  0 
0  a2  -1  0  ...  0  0 
0     ag     as  -1  ...  0     0 


0     0      0     0 


"n      «n 


+ 


F,= 


m 

-1 

0     0 

...  0 

0 

0 

Cll 

-1    0 

..  0 

0 

0 

a2 

a,  -1 

..  0 

0 

0 

0 

ttg         ag 

...  0 

0 

0 

0 

0     0 

••    On 

ftn 

,«1 


Og      ttg   ...    a„      \ 


APPLICATIONS   AND   SPECIAL   FORMS. 


183 


But  the  first  determinant  in  the  numerator  may  be  written 

0-10  0 

ai      tti      —1  0 

0      tta       ag  —  1 

0     0       tto  a. 


0 

-1 

0 

0 

«! 

0 

-1 

0 

0 

0 

a2 

— 

0 

0 

ttg 

«3 

0 

0 

0 

0 

...    0    0 

= 

...     0     0 

...     0     0 

...    0    0 

•••       On     «n 

0      0        0 


0 


...     0 

0 

...     0 

0 

...     0 

0 

...     0 

0 

•••       ttn 

Ctn 

whence  the  desired  result  is  at  once  obtained  by  substituting 
in  the  numerator  of  the  value  of  F^,  and  adding  the  deter- 
minants. 

148.  We  may,  with  the  help  of  the  preceding  article,  express 
the  value  of  the  periodic  continued  fraction 


m  + 


h  ^2  &3 


+02     +«i 


h 


as  the  quotient  of  two  determinants.      (The  *  marks  the  re- 
curring period.) 

If  we  put  X  for  the  continued  fraction,  we  have 


X  =  m  ■+■ 


h       ^2         ^3 
«i      +02      +»3 


h 


4-a2     +<^i     m+a; 


Then,  by  147, 


/     bi     62       •••       h     h     h  \ 

\m    %    0^2  •••  ttg    «!     (m-{-x)  J 


62     h 


/     62     h       " 

\ai    Og    0^3  •••  «2    ^1     (^ 

clearing  of  fractions,  and  expanding, 


/         62     h  "'       63     h     &A       ^^f        h     h 

_/     61     62       •••      h    h       \    X       (     ^^     ^2 
~"\m     Oi     tta  •••  ^2     «!     m  /  "^      \m     «!     a^ 


^2   (h) 


184 


THEORY  OF  DETERMINANTS. 


But  the  first  term  of  the   first  member  equals  the  second 
term  of  the  second  member  of  this  equation. 


x  =  ± 


(  '' 


63     62     ^1 


^     h     h       '"       h     h 

«!       ^2       «3     •••     Cts       «2       «1 


149.    Let  us  now  consider  the  ascending  continued  fraction 


as  4-   .*          a« 

—  ^                   «  ^  tti    +a2    +a3 

^-  =  ».  +  "^ 

enote  the  convergents  to  F^  by 

P          Pi          i>2                   Pn 
_,         — ,         — ,      ...      — f 

^     qi     q2        qn 

and  let  us  obtain  the  determinant  expression  for  the  nth  con- 
vergent. 

We  have  evidently 


p^  is  determined  from  the  following  equations,  which  the 
student  can  easily  deduce : 


—  O'zPt-^Pz 


=  a3 


—  ttn-a  P«-8  +  Pn-t  =  an-2 

—  an-lP«-2  +  Pn  -1  =  a^-l 

a«P»-i+i>»     =0^. 


APPLICA.TIONS  AND   SPECIAL   FORMS. 


185 


From  these  equations 


i>n  = 


1 

0 

0 

0 

0 

0 

ttl 

-^2 

1 

0 

0 

0 

0 

02 

0 

-^3 

1 

0 

0 

0 

ttg 

... 

... 

... 

... 

... 

... 

... 

0 

0 

0 

a„ 

~2 

1 

0 

an-2 

0 

0 

0 

0 

- 

-«n-l 

1 

O-n-1 

0 

0 

0 

0 

0 

<^n 

a« 

ai 

-1 

0 

0    . 

0 

0 

0 

^2 

a^ 

-1 

0    . 

0 

0 

0 

"3 

0 

«3 

-1  . 

0 

0 

0 

a«-2 

0 

0 

0    . 

•     «'»-2 

-1 

0 

a„-i 

0 

0 

0    . 

0 

ttn-l 

-1 

ttn 

0 

0 

0    . 

0 

0 

ttn 

ttl 

-1 

0 

0    . 

..    0 

0 

0.2 

^2 

-1 

0    . 

..    0 

0 

as 

0 

«3 

-1  . 

..     0 

0 

... 

... 

... 

...    . 



... 

a«-i 

0 

0 

0    . 

••     «n-l 

-1 

O-n 

0 

0 

0    . 

..     0 

dn 

ai 

-1 

0 

0    . 

.     0 

0 

0 

a^ 

-1 

0    . 

■•     0 

0 

0 

0 

^3 

-1  . 

.     0 

0 

... 

... 

... 

...    . 



... 

0 

0 

0 

0    . 

••     ttn-l 

-1 

0 

0 

0 

0    . 

.     0 

a„ 

Pn 

150.    The  numerator  and  denominator  of    -  can  be  trans- 

formed   into   continuants,    and   thus   the   fraction   F^   can  be 
transformed  into  a  descending  continued  fraction,  as  follows : 


186 


THEORY   OF  DETERMINANTS. 


Multiply  the  last  row  of  p„  by  (v-u  and  subti-act  from  it  the 
(n  — l)th  row  multiplied  by  a„ ;  then  multiply  the  (n— l)th 
row  by  a„_2,  and  subtract  from  it  the  (w  — 2)th  row  multiplied 
by  a„_i ;  and  so  on.     Then 


«! 

-1 

0 

0 

0 

agai+ag 

—  ai 

0 

0 

—  agoia   ^so^j+as 

—  ou 

0 

0 

0 

0 

0 

0 

0 

0 

Pn  = 


—  a„_2a„_i   an_ia„_2-|-a„_i    —  a„_2 


a„_ia„_2a„_3 


a^a^ai 


Similarly, 


gn  = 


a,         -1 

0 

0   ... 

0 

0 

0 

—  aittg  asttj  +  aa 

— ttl 

0    ... 

0 

0 

0 

0           —  tta^s 

«3a2-f-a3 

— ttg  ... 

0 

0 

0  . 

... 

... 



... 

... 

... 

0            0 

0 

0    ... 

a«-i 

a«-ia»-2 
+  an-i 

—  an-2 

0            0 

0* 

0    ... 

0 

-««-!«« 

an-ia»-2an-3 


a2ai 


Whence,  by  144, 


Ol 


a^a^ 


Ct2«l«3 


Ol    --a2ai+a2      —  asOg+ag 


5'« 

• 

an-2an-3a^-l 

a„_ia„_2an 

—  «n-ia„-2+an_i 

—  ««««-! +an 

the  descending  fraction  sought. 


APPLICATIONS   AND   SPECIAL  FORMS. 


187 


Alternants, 
151.    Consider  the  determinant 

A  =      1      «!      Of/      •••      ( 


1     ttg     a^ 
1     a„    a„2 


as 


and  the  product 


P  =  (a2-ai)(a3-ai)(a4-ai)   ...   K-ai) 

X  (as— «2)(a4— «2)   •••   («n— 02) 
X  (a^-as)   •••   (a^-ag) 

X  (a^-a„_i) 

of  the  -  (71— 1)  differences  of  the  n  different  quantities  involved 

in  A.  This  product  is  called  the  difference  product  of  the  n 
quantities  ai,  ag,  •••  a„,  and  for  it  the  notation  ^*(ai,  ag,  ag,  •••  a„) 
has  been  adopted. 

The  reader  will  remember  that  the  square  of  the  difference 
product  was  denoted  by  ^(ai,  as,  •••  a„),  and  thus  the  difference 
product  itself  is  very  appropriately  designated  by  C*(ai,  ag,  •••  a„). 

We  shall  now  show  that 


A   =P  =  CK«15   «2,    --J   ttn)' 


(1) 


If  in  A  we  put  a^  =  a„^  A  vanishes ;  hence  A  is  divisible  by 
each  factor  of  P,  and  hence  by  P.     Again,  A  and  P  are  each 

n 
polynomials  of  degree  -  (n  — 1),  and  therefore 

A  =  \CH«15   «2,   «3?    ••-,«„), 

where  A.  is  a  factor  independent  of  ai,  ag?  •**  <*»*•      From  the 
special  case 


188 


THEORY   OF   DETERMINANTS. 


(a^-a,)  (ccg-^i)  (a3-«2)> 


we  see  that  X=  1,  and  thus  the  truth  of  (1)  is  established.* 

152.  A  of  the  preceding  article  is  evidently  an  alternating 
function ;  for  the  interchange  of  a^  and  a„  amounts  to  an 
interchange  of  two  rows  in  the  determinant,  and  hence  changes 
its  sign.  A  is  accordingly  an  Alternant.  In  general,  an  alter- 
nant is  a  determinant  in  which  each  element  of  the  first  row  is 
a  function  of  £Ci,  the  corresponding  elements  of  the  second  row 
the  same  functions  of  x^,  and  so  on.     Thus 

/»K)] 


A  = 

fiixO   Mx,)  ...  M^,) 
Mx,)   L{x^  -  f.{x,) 

Mx,d  Mx:)  ...  Ux^) 

■^AlMx,),  Ux,) 

is  an  alternant. 

153.    We  can  easily  show  that 

A  = 

1    fx{x,)    Mx,)  ...  u,{x,) 

1        fl(X2)         A{X2)     '•'    fn-l(x,) 

=   H^(Xi,    X, 

1         /iW        /sW    -    fn 

-l{Xn) 

•   x^), 


where  fr(x)  is  a  function  of  the  rth  degree  in  x,  and  A.  is  the 
product  of  the  coefficients  of  the  terms  of  highest  degree  in 
the  several  functions.  For  subtracting  the  first  column  mul- 
tiplied by  the  proper  number  from  the  second,  we  reduce  the 
elements  of  the  second  column  to  piX^,  piX2,  p^x.^,  ...  PiX,^. 
Then  subtracting  the  sum  of  the  first  and  second  columns,  each 
multiplied  by  the  proper  number,  from  the  third  column,  the 
elements  of  this  column  become  ^2^/?  ^2^2'*)  •••  P2^n^.  Pro- 
ceeding in  this  way,  we  see  that  finally 

A  =  \tj{xi,  ^2,  •••  a;„). 


See  also  examples  6  and  7,  page  37. 


ArPLICATIONS   AND    SPECIAL   FORMS. 


189 


where  ^  =  Pi  •  i>2  •  •  •  Pn* 

For  an  example,  putting 

we  have 
A 


1        /iW         f2M     ...    f^_,(x{) 

1    Mx,)    Mx,)  ...  f,,_,(x,) 

1-        /l(^n)         /.(^n)      -    /n-l(^«) 


(71-1)!  (n-2)!  ...  2!' 


154.  Every  alternant  whose  elements  are  rational  integral 
functions  of  Xi,  X2,  •••  »„,  is  divisible  by  t}{xi^  iCg?  ^31  •••  ^n)» 
and  the  quotient  is  a  symmetric  function  of  the  variables. 
For  the  alternant  vanishes  if  Xt  =  Xj,,  and  hence  is  divisible  by 
Xi  —  x^,  and  thus  by  CH^n  ^2?  •••  ^n).  '^^^  quotient  must  be  ^ 
symmetric  function,  for  the  interchange  of  Xi  and  Xj,  changes 
the  sign  of  both  dividend  and  divisor ;  therefore  the  sign  of  the 
quotient  remains  unchanged  upon  the  interchange  of  two  of 
the  variables,  and  is  accordingly  a  symmetric  function.  We 
shall  now  actually  perform  the  division  just  considered.  Alter- 
nants whose  functions  are  powers  of  the  variables  are  called 
simple  alternants^  and  are  of  frequent  occurrence.  We  proceed 
first  to  the  discussion  of  simple  alternants. 

155.  The  quotient 


1    X, 

...  arr'^i' 

1    X, 

"'xr'x2' 



1  x^ 

-'X^--'X^^ 

A(Xj\x2,  Xs^"'Xl_-i,X„^) 


-f-  i  {X,,  X,,  ...  X^)  =  ^^^-_____^ 


may  be  developed  as  follows  : 

Expand  the  dividend  A  in  terms  of  the  elements  of  the  last 
column,  and  we  obtain 


190 


THEORY   OF  DETERMINANTS. 


da^i'  die/  dx/  dxj 

Now,  it  is  evident  that  each  of  the  minors  in  this  expansion 
is  a  difference  product. 
Thus 


dA 
dxj 


^(-1) 


-L       3/j        flJj 


,n-2 


1     «,_i  a^_i  •••  x^i! 
1      OJ,.^!  a^^i   •••   x^^i 


1      a7„      £C„ 


icr 


Substituting  in  (1)  the  values  of  the  minors  as  found  from 
(2),  and  dividing  both  members  of  (1)  by  l^{xi,  iCg,  •••  a;„),  we 
have  a  series  of  terms,  of  which 

(-!)»+"  a;/ 

lx^-X,){x„_-^-X,)   ...   {Xr+i  —  Xr)  (X,-X^_i)   •"  (x,  —  X2){X,-Xi) 

is  the  type.     Thus  we  find 

C-{Xi,  X2,  ...,  x„) 
r=i  {x^-Xr)(x^^i~x,)  ...  (a;,+i-a;,)(a;,-a;,_i)  ...  {x^-x,)  ' 


or 


a^i' 


4....  + 


(X,- 

-a;,)  (»,- 

-w„ 

-.) 

••(^1 

-X,) 

+ 

x,^ 

(».- 

-a!„)(X2- 

-a;. 

-.)• 

..(0^- 

-X3)  (asj- 

-a;,) 

^L. 

(aJ„-2-««)(a;„_2-a?n-i)(»«-2-»n-3)  •••  (a;«-2-aa) 


APPLICATIONS   AND   SPECIAL  FOKMS. 


191 


+ ^ 

For  an  illustration,  we  have 


1     2     16 

-f- 

1     3     81 

1     5    625 

2 

4 

3 

9 

5 

25 

16 


+ 


81 


625 


=  69. 


(2-5)  (2-3)        (3-5)  (3-2)        (5-3)  (5-2) 

156.    With  the  help  of  the  preceding  article  we  may  reduce 
the  quotient 

^i(a;i,  ajg,  •••  «„) 

to  the  sum  of  two  similar  and  simpler  quotients,  as  follows : 
Since 


A{X^,  X},  Xi,    ...   <:f,   V)   -  ^n^i^X-,  ^2, 


K'-l.   ^n'-') 


1     x^     x^ 

1      X2      xi 


1      X^_i    af„_2  •«.       flJ^-l         ^n-l(^n-l  —  ^n) 


1    x^ 


XI 


0 


(1) 


we  have,  after  dividing  both  members  of  (1)  by  l^{xi,  x^,  •••  x^) 
in  accordance  with  155,  and  striking  out  the  factor  common 
to  numerator  and  denominator  of  each  term  in  the  second 
member, 

A{x,\  x},  xj,  ...,  x^-l  x,^)      x,A{x,\  x,\  x,\  ...  <:f,  x^') 


192 


THEOBY   OF   DETERJVaNANTS. 


q-l 


^i 


?-l 


+  ...  + ^ 

K-i-aJ»-2)  {x^-\-Xn-z)  •••  (a^n-i-a^i) 

But  the  sum  in   the   second   member  of   this   equation   is, 
by  155, 


1       Xi        Xi 

J.  Xo  Xo 


x^-'^     x^-'^ 


1       ^n-1  Xn-\   ••• 

Transposing,  we  have 


/v.n-3 


^H-1 


-^  lK^i'>^2'>  •••ja;„_i). 


^H^i'  ^2,  •••  a;„)  ^K^i5  ^2,  •••  x^) 


+ 


A.{Xi  ,  a72 ,  i^3  , 


a^^Il) 


^n-3     ^?-r 


which  is  the  desired  reduction.     For  example, 


\     X    a? 

1     a;     «2 

1    y   f 

1    y    f 

1     a.-2 

1   2   ^ 

1     ^     2^ 

1 

1        2/2 

^i(a;,  2/,  2!)  4Ha^»  ^5  ^)       ^H^»  2/) 


=  a;  +  y  +  2  =  2a;. 


I     X     x'^ 

1     X    a^ 

1     2/    2/* 

1     2/    f 

1     a!« 

1     0     ;2^ 

1       «       2? 

, 

1     2/» 

The  student  may  show 


=  '^a^  +  'Z^y-^-xyz, 


1 

X 

a^ 

1 

y 

f 

1 

z 

z" 

^H^5  2/5  2;) 


APPLICATIONS   AND   SPECIAL  FORMS.  193 


-^x^  +  %a^y  +  5a^2/2  _^  ^^^^^ 


1 

X     x' 

1 
1 

y   f 

z     z' 

4*  (^,  y,  z) 


157.  Since  every  term  of  CK^i'  ^2^  •••  ^n)  contains  a  permu- 
tation of  all  the  powers  of  the  variables  from  1  to  71  —  1 ,  each 

term  is  of  the  -(n  — l)th  degree.      Similarl}-,  every  term  of 

A(x,\  x,\  xi,  ...,  <:f,  0^,0    is  of   degree   iliZliliZLlA)  +  g. 

Hence  every  term  of 

f;^^A{x,\x,\xi,  '".xlzlx^)      , 

i\x,,  X2,   '",  X,) 

is  of  the  (q  —  n -\- l)th.  degree,  as  is  illustrated  in  the  exam- 
ples of  the  preceding  article.  We  shall  now  show  that  every 
possible  term  of  the  (g  — n4-l)th  degree  in  the  variables  is 
found  in  Q,  and  that  every  such  term  is  positive.  That  is  to 
say,  the  quotient  Q  is  the  complete  symmetric  function  of  degree 
(g- 71  4-1)  of  a^i,  X2,  •••,  x^. 
Such  a  term  of  Q  is 

1  =i  Xi  X2  x^  '•'  X n_2 ar„_i x^. 

By  successively  applying  156,  we  develop  Q  so  that  the 
terms  containing  ic„,  aj^a^^i,  x^x^^_ix\,^2i  etc.,  are  at  once 
distinguished.     In  the  first  place, 

^^«,a;2^a^3^'••,<:i,a^r-l)   .  x.,A(x,\x2\x,\  •^-.xZzlxlzl) 
t}{xi,  X2,  ...,  x^__{)  ^i(a;i,  X2,  •..,  a;„_i) 

^aiA(x,',X2\xi,...,xl-lxt_\)  ^     ^  xr''A(x^,X2\xir';x:izixZz]) 


The  second  term, 

X^Ai^Xi  ,  X2  ,  X^^   '  • »  ,  a?;i,„2^  ^ra    1  ) 


=   «1, 


194  THEORY   OF   DETERMINANTS. 

contains  the  first  power  of  x^ ;  hence  we  must  look  for  T  in 
Qi.     Applying  156  to  Qi,  as  before,  we  have 


^-1^(^1  ^  ^2  ?  ^3  ^   "'^  ^n-3?  ^n-2)     I 
t    (^li  ^21    •*•?  ^n-2) 


a^n:r'^(«^l^  ^2\  «^3',   -,  ^n-l)   j^^ 


-'} 


4  (a?!,  a;2i  *  *  *  5  ^n  2) 

In  this  expansion  we  must  look  for  T  in  the  third  term 

% 

^n^n-\-^\^\  ^  ^2  ?  ^3  ^  •••?  ^n-  3?  ^n    '2)    __   Q 
t    (^1?  ^*2?    '"j  ^n-i) 

Q2  may  be  expanded  as  before ;  continuing  in  this  way,  we 
finally  obtain  the  term 

X  aP     x^      ...T«  ^W^2')  . 

^^{X^X2) 

for  the  coefficient  of  a7„«l_ia;^„-2  •••  a^s^  contains  only  a^i  and  a?2, 
and  is  of  the  third  degree.  Upon  performing  the  division,  and 
multiplying,  one  of  the  terms  is  T.  Since  T  is  any  term,  the 
proposition  is  established. 

Employing  the  notation  H^  for  the  complete  symmetric 
function  of  the  rth  degree,  we  may  write  the  result  of  the 
present  article 

A(x,\x,\x,\....xlzlx,^,)_  jj        (^    X    ...  x\ 

or  simply  Hq^^+i* 

For  illustrations  the  student  may  refer  to  the  examples  in 
the  preceding  article. 

«  With  this  notation,  Hq  =  1,  ZT.,  =  0. 


APPLICATIONS   AND   SPECIAL  FOKMS. 


195 


Again, 


1  X  x^  x^ 

1  y  y^  f 

1  z  ^  ^ 

1  t  f  f 


=  iJ^  =  -^x*  +  20^2/  +  Ix^y^  +  ^xyzt. 


l'-{x,  y,  z,  t) 

158.  From  the  two  preceding  articles  we  have  at  once 

H^{X^,X^,  "•,X^)  =  X^H,_.l(Xi,X2,  '••,X^)-\-Hr(Xi,X.,,  •••,i»«_i). 

(1) 

Whence  we  readily  obtain 

Hr.i{x„x.>,  '•■,x^+i)  =  x^^^H,_ci{x^,x^,"',x^^^)-^H,_^{x^,X2, ...,  x^y, 
Substituting  in  (1), 

-"r(^l?  ^2»   "'-i  ^n)  =  ^nL-"r-l(^l5  ^2>    '**?  ^n+l) 

- x„+i^,_2(a;i,  a^a,  •••,  Xn+i)~\+H,{x^,  x^,  ...,  a;„_i).         (2) 
Similarly, 

XZy(i»l,  a;2»   •••>  ^n— l^«+l)  =  ^n+lL-"r-l(^l?  ^25   '"^  ^n+l) 

From  (2)  and  (3), 

Hri^l-,  a^2,   •••  ^n)  —^ri^l,  X^,   '"  X^_iX^+i) 

=  (^n-  Xn+l)Hr-l(Xi,  X2,    '  -  •  iK„+i)  .  (4) 

159.  If  any  alternant  whose  elements  are  powers  (simple 
alternant)  be  divided  by  the  difference  product  of  its  variables, 
the  result  is  expressible  as  a  determinant  whose  elements  are 
complete  symmetric  functions  of  the  variables.     That  is  to  say, 


Ajx'^,  xi, ...  0^::) 


-fftt-n+l   H^-n+l 


H^- 


-n+l 


196 


THEORY   OF   DETERMINANTS. 


This  may  be  proved  as  follows.  For  brevity  we  employ 
determinants  of  the  third  order,  but  the  method  applies,  of 
course,  to  determinants  of  any  order.*     In  the  alternant 


Xi 


x^      x^ 


(1) 


subtract  the  first  row  from  each  of  the  other  two,  then  remove 
the  factors  {x^  —  Xi) ,  (x2  —  Xi) .  Afterward  subtract  the  second 
row  from  the  third,  and  remove  the  factor  ccg  — a^g?  employing 
equation  (4) ,  158.     The  result  is 


A{xi,  Xj,  x^)  _ 


ll^{X,)  '  ll^(X,)  ffy(X,) 

H^_.2{X^,X2,^5)     H^-2{X^,X2,X^)     IIy-2{Xi,X2,X^) 


The  determinant  on  the  right  we  now  transform  as  follows. 
Add  the  second  row,  multiplied  by  ajg,  to  the  first,  employing 
equation  (1),  158,  and  obtain  the  determinant 

IIa{Xi,  X2)  H^i^lt  i»2)  Hy{X^^  Xo) 

IIa-2{Xi,  X2,  Xs)      Hp-2(Xi,  X2,  ajg)      Hy-2{X^,  X2,  x-i) 

Now  add  the  third  row,  multiplied  by  ajg,  to  the  second, 
again  employing  (1)  of  158;  finally,  add  the  second  row, 
multiplied  by  iCg,  to  the  first. 

We  then  obtain 


^i(a;i,  X2,  x^) 


Ha(x^,X2,Xs)  B'p(Xi,X2,Xs)  Hy(x^,X2,Xs) 

Ha-l(Xi,X2,Xs)     n^-l{X^,X2,X^)     Hy-l{Xi,X2,X^) 
Ha-2(Xi,X2,Xs)    H^.2{Xi,X2,X^)    Hy-2{Xi,  X2,  X^) 


as  was  to  be  shown.     For  an  example, 


*  The  mode  of  proof  here  given  is  due  to  Mr.  0.  H.  Mitchell,  American 
Journal  of  Mathematics,  Vol.  IV.,  page  344. 


APPLICATIONS   AND    SPECIAL   EOHMS. 


197 


a 

a' 

a' 

=  abc 

h 

b' 

b' 

c 

c' 

(^ 

=  abc^^{a,b,c) 


=  abc^^{a,b,c) 


1  a^  a* 
1  b^  b' 
1     c"     c^ 


HQ{a,b,c)     IIs(a,b,c)     H^{a,b,c) 


-Hi     ^2 


=  abGt^{a,  b,  c)  |  Sa^  +  ^ab     ^a"  +  %a-b  +  %abc 


=  a6c^i(«»  ^  c)  I  -  2a5      -  Sa^^  -  2  2a6c 
1      2a  2a'  +  2a6 


=^abc^^{a,bj  c) 


—  2a6     2a6c 
2a      —  2a6 


160.    Form  the  product 


an 

ai2    • 

•       «1« 

X 

xr' 

^-»  . 

•    a^i 

1 

«21 

^22       • 

•       «2n 

a,r' 

•     X2 

1 

... 



.        ... 

... 

...     . 

.    ... 

.. 

a„i 

«n2      •• 
_i 

.    /I... 

i»,"-2    . 

•       ^n 

1 

changing  the  columns  of  the  first  determinant  into  rows  before 
multiplying.     If  we  put 

f,{x)  =  a^.x''-^  +  a^.x''-"-  H \-  a^_^,x  +  a„„ 


we  find 


p=(_l)l(-^>|a,J^K^1.^2,-,aJn) 


/l(»2)         f2(X2)        •••       fn{X2) 
/l(^»)        /2(a'n)      •••       /«(a;„) 


193 

If  now 
we  must  have 

laiJ  = 
m-1 


/ 


TRBOBT   OF   DETERMLNANTS. 


Mx.)^^x.-y;)'-\ 


1  (:T)i-y^^  Wy-y^y  [vy-y^y 


1  (7')^-^")  {^ty-y^^y  (Vy-y^y  -  (-^»)"-^ 


where 


But  this  last  determinant  evidently  equals 

where  K  is  the  product  of  all  the  binomial  coefficients  of  order 
w  —  1.     We  have,  accordingly, 

(aJi-2/i)"-'      (a^i-2/2)""'     -     (aJi-2/n)""' 
(«2-2/i)""'      (^^2-2/2)""'     •••     (a?2-2/n)"~' 

(x^-y^y-'    {x^-y^y-^   -    K-2/n)"-' 

=  /r^*(^l>  «'2,  a?3,  •••,  a^n)  ^*(2/l»  2/25  2/3,   •••,  2^n)  • 

If  now  x^  —  y^y  we  have  C(^i>  ^2?  ^s>  •••^^n)  in  the  form  of  a 
determinant. 

161.    Siippose  now   that    aj,  ag,  •••,a„    are  the  roots  of   an 
equation 

f(x)  =  0.  (1) 


APPLICATIONS   AND   SPECIAL  FORMS. 


199 


Then  ^^(ai,  a^,  ag,  •••,  a„)  is  the  product  of  the  differences  of 
the  roots  of  (1).     Square  this  determinant,  obtaining 


^  (ttj,  as,  •  •  • ,  an)  = 


1  +  1    +-  +  1. 

«!       +CL2      H f-a„ 

a/     +ai     +...+a,f 


«!  +a2  H f-an 

ai^  +  a^^  +  .-.+a^f 


tti 


n-l 


af      +a2'*      + 


+  a 


+  a„" 


+  a2"-i+...+a,'*-i 


-"+'  +  02"+'  + 


+  a 


n+l 


a,'--'+ai--'  + 


H-a„^' 


So 

Sl 

S2 

Sn-1 

Si 

S2 

S3 

s» 

S2 

S3 

s. 

^n+l 

... 

... 

... 

... 

s«- 

iSn 

Sn+1 

S2»-2 

where,  as  usual, 


S^  =  ai*-  +  a2'*+  •••  +a,r. 


162.  The  preceding  article  gives  us  an  expression  for  the 
square  of  the  differences  of  the  roots  in  terms  of  s^.  We  can 
also  readily  obtain  an  expression  for  the  sum  of  the  squares 
of  the  differences  in  terms  of  s<  as  follows. 

We  have 


by  58. 


1     1 

P   y 


Sl 


Si       So 


=  2(a-^y 


163.    We  shall  conclude  our  discussion  of  alternants  with  a 
theorem  on  the  reduction  of  alternating  functions  to  alternants.* 


*  "  Reduction  of  Alternating  Functions  to  Alternants,"  Wm.  Woolsey 
Johnson,  American  Journal  of  Mathematics,  Vol.  VII.,  page  345. 


200 


THEORY   OF   DETERMINANTS. 


(1) 


Any  function  of  the  form 

4>i{a,  bed  •••  I)      4>2(iii  bed  •••  I)    •••    <^„(a,  bed  •••  I) 
<f>i{b,  aed  '"  I)      </)2(6,  acd  •••  Z)    •••    <j>n(b,  aed  "- 1) 

r 
<f>i{l,  abc  •••  k)     cf>2{l,  abc  •••  A:)    •••    ^^(Z,  abc  •••  k) 

is  evidently  an  alternating  function  of  a,b,c,  "•  I,  if 

<^(ci,  6ccZ  •••  Z) 


denotes  a  function  of  the  n  quantities  a^b,c,'"l,  which  is 
symmetrical  with  respect  to  all  the  quantities  except  a.  If 
each  element  of  this  determinant  contains  only  the  leading 
letter,  (1)  becomes 


Mb) 


M<^) 
Mb) 


/3(«) 

Mb) 


M')     Ml)     Ml) 


Mo) 
Mb) 

Ml) 


(2) 


an  alternant,   which  we  represent,  as  usual,  by  its  principal 
term, 

[/l(«),/2W,/3(c),-/n(0].  (3) 

Now,  if  the  principal  term  of  (1)  can  be  separated  into  parts 
of  the  form  (3),  then  the  given  alternating  function  (1)  is 
equal  to  the  sum  of  the  alternants  represented  by  these  partial 
terms.  This  is  proved  as  follows.  Since  an  interchange  of 
two  rows  of  (1)  is  equivalent  to  an  interchange  of  the  corre- 
sponding letters,  any  term  of  (1)  can  be  obtained  from  the 
principal  term  by  a  suitable  transposition  of  the  letters,  and, 
similarly,  the  corresponding  term  in  each  of  the  alternants  may 
be  derived  from  its  principal  term  by  the  same  transposition 
of  the  letters ;  hence  every  term  in  the  expansion  of  (1)  is 
equal  to  the  sum  of  the  corresponding  terms  in  the  expansion 
of  the  alternants. 


APPLICATIONS   AND   SPECIAL   FORMS. 


201 


Accordingly,  if  a  determinant  of  the  form  (1)  is  expressed, 
as  usual,  by  writing  its  principal  term  in  (),  with  commas 
between  the  elements,  we  may  erase  the  commas,  and  treat  the 
expression  within  the  (  )  as  an  ordinary  algebraic  quantit}'. 

Thus, 


=  A{bcd,  1,  c,  d^)  =^«  6,  r,  d^)  =  t,^{a,  b,c,  d). 


bed 

1 

a 

a' 

cda 

1 

b 

W 

dab 

1 

c 

(? 

abc 

1 

d 

d' 

Again, 


1  b^  +  (?  a'  +  bc 
1  c2-fa2  b^  +  ca 
1     a'  +  b^    c'  +  ab 


=^(l,c2  +  a2,  c2  +  a6) 


A{a\  b\  c')  -\-A(a\  b\  c')  +A{a,  b,  c')  +A{a\  6,  c«) 
-  A  {a\  6,  c^)  =  _  (a  -h  6  +  c)  C*  (a,  b,c).      ' 


Functional  Determinants. 

164.   Consider  the  following  n  functions  of  the  n  independent 
variables  iCi,  X21  •••  x^. 


2/i=/i(a?i,a;2,  "',x^) 


(1) 


These  functions  will  be  independent  if  for  every  set  of  values 
of  yi,y2,"'yn  equations  (1)  determine  one  or  more  sets  of 
values  of  a^i,  ajg,  •••  a;„,  so  that  these  latter  variables  can  in  their 
turn  be  considered  as  functions  of  the  n  independent  variables 

yi,y2,"'yn' 


202  THEORY   OF  DETERMINANTS. 

Differentiating  equations  (1),  we  have 

dy,  =  ^'dx,  +  pdx2  +  .-.  +  ^dx^ 


hXi 
hXi 


dy2=^'dx,-\-^dx2  + 


Sx^ 

SX2 


Sx^ 


dy, 


^^Adx^+^^dx2  +  •••  +^d.x„ 


hxi 


Sxo 


6x„ 


(2) 


Regarding  equations  (2)  as  a  system  of  equations  for  deter- 
mining dxi,  dx2,  •"  dXn,  the  determinant  of  this  system 


S?/i 

S«2 

S.Vi 

••,2/.) 

%2 

8x, 

%2 

8& 

8& 

%, 

8a;, 

8x, 

■     8*„ 

is  called  the  Jacobian  of  the  given  functions  yi^  2/2?  *  *  •  Vn' 
Or,  in  other  words,  the  Jacobian  of  a  set  of  n  functions,  each 
of  n  variables,  is  the  determinant  |A:i„l,  in  which  the  element 
kpg  is  the  first  derivative  of  the  pth  function  with  respect  to  the 
^th  variable.     Thus,  given 

yi  =  az^  -\-2  hzt  +  ci^,     2/2  =  ^i^!^  +  2  hiZt  +  Cif. 

The  Jacobian 


az  +  6i      hz  -\-  ct 
a^z  +  &i^     h^z  4-  Ci? 

_4    1 

4 

0       0      1 

=  4 

^2     _2«       22 

. 

"^, 

2/1     hz  -\-ct     c 

a     6       c 

^' 

Vi    h,z  H 

-  Ci«      Ci 

a,    « 

'1         Cj 

APPLICA.TIONS  AND   SPECIAL   FORMS. 


203 


165.    If  the  functions  2/1,2/2,  •••  2/n  ^^^  iiot  independent,  but 
are  connected  by  a  relation 


<^  (2/15  2/2,  •••2/n)  =  0, 

the  Jacobian  vanishes. 

From  (3)  we  have,  by  differentiating, 


(3) 


%l   ^Xl     82/2   ^^1 

82/1    80^2      82/2    8a;2 


82/n     S«^i 

H-  ^  .  ^"  =  0 
S2/«     80^2 


(4) 


§i  .  ?^i  _i_  ^  .  5^2  ^ ...  _i_  ^  .  ^  =  0 

82/1     8a;«      82/2     8aj,  8y^     Sx^ 


From  their  mode  of  formation,  .equations  (4)  are  simulta- 
neous.    Hence  the  determinant  of  the  system  vanishes  by  77  ; 

^  =  0. 

We  shall  show  presentl}^  that  if  the  Jacobian  of  a  set  of 
functions  vanishes,  the  functions  are  not  independent. 


166.    The  Jacobian  of  the  implicit  functions 

Fi  (a^i,  X2,'"  cc„,  2/1, 2/2,  •••  2/n)  =  0  ^ 
F2  (a^i,  X2,  •••  a;„,  2/1, 2/2?  •  •  •  2/n)  =  0 


(5) 


Fn(Xi,  »2,   •••  X^,  2/1?  2/2J  •••  2/n)  =  0 


is  found  as  follows. 
Equations  (5)  yield 

_SF\^§T\  .  S^i_^S5    8^2  ^  ...  ^  55-  .  %2 
Sxj,      82/1     Bxj,     82/2     Sx^  Sy^    Bx^ 

(1,  A;=l,  2, 


(6) 


n). 


204  THEORY   OF  DETERMINANTS. 

Using  equation  (6),  we  find  the  product  of 


to  be 


8F,    8F, 

SF^ 

and 

%2 

8x, 

Byn 

'"     8x, 

8F2    SF, 

%1         %2 

8F, 

8x, 

5^2 

6X2 

8x2 

BF^    8F„ 

Byi    %2 

SF^ 

Syi 

8x„ 

%2 

Byn 

'"     8x^ 

(-1)" 

SF,    8F, 

... 

8F, 
8x^ 

• 

SF^    8F2 
8x1     8x2 

... 

8F2 

Si^n        S^n 

BF^ 

8x^ 

.8x2 

... 

Sx^ 

Whence 


j^Hyi.y2,'-yn)  ^.       8(j^i,j^„-..i^0  .  bjf^f^^-'F^) 

8{xi,  X2,  •••  x^)  8{xi,  X2,  —  x„)  '  8(2/1,  2/2»  •••  2/n) 


(7) 


If  in  (7)  we  put  n=  1,  we  get 


8F\_8F\  dyi 
8x1      8yi  dxi 


a  well-known  formula. 


167.    If  in  equations  (5)  we  consider  ajj,  ajg?  •"  x^  as  func- 
tions of  yi,  2/2>  •••  2/rn  W6  obtain,  as  above, 

r^yS{F„F2r"F,)  ^8iF,,F2r"F,)  ^8(x,,X2,'"X^)         ,g. 
8(2/1,2/2,  •••yn)      S(a;i,a;2,  •••»„)      8(2^1,2^2,  •••  2/«)* 


(9) 


APPLICATIONS  AND  SPECIAL  FOBMS.  205 

From  (7)  and  (8), 

8(a;i,  x^,  ...  x„)       8(2/1,  2/2,  ...  y^) 

168.    Again,  having  given  the  n  -{-p  functions, 

2^1  (xi,  x^,  ...  a;,,,  2/1,  2/2,  •••  2/n+p)  =  0 
F^ix^,  %,  •••  a?„,  yi,  2/2,  —  2/«+;,)  =  0 

Fn-^{xi,  02,  —  a;„,  2/1,  ^2,  •••  2/n+i>)  =  0 
The  Jacobian 

J-^  8(2/1,   ^2,    —   Vn) 

o(aJi,  a;2,  •••  x^) 

of  the  first  w  of  these  functions  is  found  as  follows.     Differen- 
tiating equations  (10),  we  find 


(10) 


8x,       Sy,  Sx,  "^  By,  Sx,  "^       "^  Sy„+,    So;, 


(6) 


(*  = 

1,2,... 

7i+i>; 

A:=l,2, 

...n) 

Now  multipl}'  together 

A  = 

8i^i        8i^i 
82/1        82/2 

8i^i 

S2/n+p 

X 

S2/1 

%2 

8a;i     •*' 

S2/n 

8a;i 

8F2        8i<T, 
%i        82/2 

8F2 

S2/1 

8^2 

^2/2 
8a;2 

S2/n 
80^2 

%!              8^2 

SF^^P 
^Vr^p 

%1 

8cc„ 

%2 

8a^n       *'* 

82/n 

first  writing  J"  as  a  determinant  of  order  w  -j-p?  thus  : 


206 


THEOBY  OF   DETERMINANTS. 


8^1 
0 


8xi      8xi 

BX2         SX2 


8x,      8x^ 
0         1 


0         0 


Sxi 


SX2 


Calling  the  product  P,  we  have 


p^(-iy 


8a;i 

SaJi 
8P„. 


8Pi         8Fi 


8a;„ 

8y„+i 

s-f;+. 

8-F-n+, 

Sa^i 


8Pi 


n+p 


SP„ 


=  (-!)% 


8a;„      82/„+i 

S(Pi,   ^21***  ^^i+p) 


^Vn+p 


6  (a?!,  a;2»    •  •  •   ^n»     2/n+l»  2/n+2>    •  •  •    Vn+p) 

since,  by  equation  (6)  for  A;^n,  the  element  Ui^  of  P  is ? ; 


and  for  A;  >  n, 


8^ 


gajfc 


We  have,  accordingly. 


-=£• 


169.    Suppose  equations  (5)  yield  upon  solution 
2/1  =  </>i(a?i,  3^2,  •••,a;„). 


(0 


APPLICATIONS   AND   SPECIAL   FORMS. 


207 


Solve  (c)  for  x^,  and  substitute  this  value  of  Xi  in  the 
remaining  ii  —  l  equations  ;  then  2/2, 2/3,  •••  y^  become  functions 
of  ?/i,  a?2,  •••  ic„.     Thus 

2/2=  <^2(2/i,  372,  •••ajn).  W 

Solve  (c?)  for  »2?  substitute  the  result  in  the  remaining  n  —  2 
equations  ;  then  2/3, 2/4,  •••  2/n  become  functions  of 


2/l>  ^29  ^39    *••?  ^n* 

Thus  2/3  =  <^3(.yi,  2/2,  ^3»  •••,  a^«). 

Solve  (e)  for  0^3,  substitute  as  before  ;  and  so  on. 
We  obtain  the  equations 

2/1  -  c}>i(Xi,X2,  •••«„)  =0 

y2  —  4>2{yuX2,'"X^)  =0 

3/3-  <^3(2/l9  2/2,  3^3,  •••«?„)  =0 

2/«  -  <^n(2/l?  2/2,   •••  2/n-l,      •••  a„)  =  0 


By  166, 

j^B(y,, 

2/2,  •• 

•^n) 

=  (-1)" 

8{x,,x,,'..y^) 

hXi          8X2 

^01    ...          Bcfji 

8x,    "       8x^ 

1 

0 

0 

0      -^2 
Sx, 

Oc/j2                      002 

6x,  '"       g^ 

802 

%1 

1 

0 

0          0 

Hs        .            S03 

3aJ3             K 

803 

S2/1 

803 

82/2 

1 

0          0 

0   ... -% 

S0n 
8^1 

80n 
%2 

B<f>n 
82/3 

_  8<^,  _  802  . 
8i»i     8x2 

803 

8x^ 

(e) 


(11) 


208  THEORY   OF  DETERMINANTS. 

That  is  to  say,  the  Jacohian  of  a  set  of  functions  2/1, 2/2?  •••  2/n» 
each  of  n  independent  variables  x^,  0^2,  •••  a;„,  is  expressible  as  a 
product  of  n  differential  coefficients  of  the  functions  <f>i,  <^2>  *  •  •  S^n* 
where  </)^  is  a  function  of  ?/i,  3/0,  •••  2/r-i»  ^n  •••  ^n- 

170.  The  result  just  obtained  may  be  employed  to  show  that 
if  the  Jacobian  of  a  set  of  functions  vanishes,  the  functions  are 
not  independent. 

For,  if  r=5ii.?&     ...     Hn. 

hxi     8x2  8x^ 

vanishes,  some  one  of  the  coefficients,  say 

where  i  has  one  of  the  values  1,  2,  •••  n.  But  if  -^  =  0,  <^< 
does  not  contain  cc^,  i.e.,  * 

Vi  =  ^iiVi^  2/2,  •••  Vi-i,  ^i+u  •••  a^n). 
Also  ?/,+i  =  <^f+i  (2/1, 2/2,  •••  2/i5  ^i+u  •••«'«). 

From  these  two  equations, 

2/i+i  =  «Ai+i (2/1^  2/2?  •••  2/t?  »<+2,  a^i+s,  •••  ^«) ; 

therefore  2/i+i  does  not  contain  x^^-^.  In  the  same  way  we  may 
show  that  2/<+2  ^^^^s  not  contain  a;^^.2,  and  so  on.  Hence, 
finally, 

2/«=  «Ah (^1^2/2,  •••2/«-i); 

or  y^  is  expressible  as  a  function  of  the  remaining  n  —  1  func- 
tions, and  hence  the  given  functions  are  not  independent. 
For  example,  if  the  given  functions  are 

(1)    u=x-\-y,       (2)  v=:x  —  z,        (3)    w  =  qi^ -\-xz  —  yz —  z^, 

1       1  2/  +  ^ 

1       0  x-z 

0     -1     x-y-2z 


APPLICATIONS   AND   SPECIAL   FORMS. 


209 


J  evidently  vanishes.  Accordingly,  (1),  (2),  (3)  are  not 
independent.  That  the  given  functions  are  not  independent  is 
easily  shown  directly  as  follows.     We  readily  obtain 

y  z=u  —  'V  —  Z.        .'.  W  =  V  {u  —  V), 

as  was  to  be  shown. 


171.   If  the  functions  y^,  y^,  •••  y^  are  the  n  partial  derivatives 

ic„) ,  the  Jacobian 


V-'    /"'    •*•    -r-'  of  a  function /(a?!,  0^2, 
hXi     bxo  bx^ 


•ff(/)  = 


sy 

ixiSxi 

sy 

ay 

sxi 

sy 

sy 

Sx^Sxi    8x^8x2 


ay 

8xi8Xn 

jy_ 

Sx28x^ 


sy 

8x^' 


is   called   the   Hessian  of    (x^,  x^,  •  •  •  x^ .      The   Hessian  is  a 
symmetrical  determinant,  since 

jy_  =  J!/L. 

SXiSxj,      Sxj^BXi 
If  the  derivatives  -^,  -—  •••  r^^    are  connected  by  an  equa- 

OXi    6X2         ox,^ 
tion,  with  constant  coefficients 

dXi  6X2  Sx^ 

the  Hessian  must  vanish. 


172.  Let  /i, /2,  "•  fn  be  n  given  functions  of  the  same 
variable  x.  Suppose  the  functions  are  connected  by  the  linear 
relation 

tti/i  +  a-,fo  +  aj,  +  ...  +  a„/^  =  0,  (1) 


210 


THEORY  OF  DETERMINANTS. 


in  which  ai,  ag,  •••a„  are  not  functions  of  x.     Differentiating 
(1)  successively  n  —  1  times,  we  have 


ai/l"       +«2/2"      +«3/3"      +-«n/u"       =0 

Eliminating  oti,  ag,  •••  ««  from  (1)  and  (2),  we  find 


(2) 


/i      fi      h 

f\      fi      fz 
■fit     fti     ftt 

Jl         J2  J& 

/'n-1     -fn—l     -fn—l 


= -D  (/!,/«/« •••/,)  =  ()• 


(3) 


The  determinant  of  (3)  has  been  called  the  Wronskian  of 
fii  f2y  •"  fn'  We  see  from  (3)  that  if  the  functions  /i,/2,  •••/» 
are  connected  b}"  a  linear  equation  of  the  form  (1),  the  Wron- 
skian vanishes. 


173.  If  we  denote  the  given  functions  by  2/1?  2/2?  •••  2/n?  and 
the  derivatives  by  2/11, 2/21?  •*•  {i'^'->  the  second  subscript  denoting 
the  derivatives),  we  may  write  (3) 


Vn 


y2 


Vn 


=  D{yi,y2iy3,  •••2/n)=o. 


Vln-i     2/2  «-l      •••        Vnn-l 

Now  y  being  any  function  of  »,  we  find 

2/"-0(yi5  2/2,2/3,  •••2/«)=  yiy   iyiy)i 
»    (2/22/)  1 

y^    (yny)i 


(yiy)n-i 

(2/22/)  n-l 
(2/n2/)n-l 


(4) 


(5) 


APPLICATIONS   AND   SPECIAL   FORMS. 


211 


in  which  the  subscript  k  of  {yiy)k  means  the  kih  derivative  of 
{yiy).  That  is  to  say,  the  Wronskian  of  2/i2/?  2/22/v  •••  2/n2/  is 
the  product  on  the  left  in  (5).  This  is  made  evident  by  notic- 
ing that  since 

(2/i2/)i  =  ViiV  +  ViV^     {yiy\  =  2/<22/  +  ^yav'  +  ViV", 

etc.,  where  y',  y",  •••,  are  the  successive  derivatives  of  ?/,  the 
determinant  on  the  right  becomes  a  sum  of  determinants,  of 
which  the  first  is  the  product  on  the  left,  and  all  the  rest 
vanish. 


174.    We  find 

dD{y^,y2,-',yn) 

Vi 

yn        —       yin-2       2/ln 

dx 

2/2 

2/21        •••       2/2n-2       2/2n 

Vn 

2/«i    —     2/»«-2    2/«« 

(-1) 


for  in  the  sum  of  determinants  which  make  up  the  derivative 
sought,  all  vanish  except  the  one  expressed  in  equation  {A) . 

175.    If  in  173  we  put  y  =  —,  the  Wronskian  on  the  right  in 


(5)  reduces  to 


2/1 


.2/1/1 


ri/2 
'2/3' 


V2/1A    V2/1/2 


'y2' 
.2/1 


\yJn-i 


D 


yA  fys' 
.2/1/1 


Now 

'2/2 


i/i~      y?     '  Wi"      yi     '  "*  \yJ~     y? 


212  THEORY  OF  DETERMINANTS. 

Then  if  we  put 

D  (?/i,  2/2)  =  2:2,  D  (2/1, 2/3)  ^z^,  '"  D  (2/1, 2/„)  =  2!n1 
we  get 

^  (2/1, 2/2,  •••  2/n)  =-;n^  fe,  23,  •••  O-  (6) 

2/1 

176.  We  shall  employ  the  result  just  obtained  to  show  that 
if  the  Wronskian  of  2/15  2/2^  •**  2/»  vanishes,  the  functions  are 
connected  by  a  linear  equation  having  constant  coefficients. 
Suppose  that  2/1  does  not  vanish,  and  since  by  hypothesis 

D{yi^y2^  •••2/n)  =  0, 
by  (6)  of  the  last  article  we  must  also  have 

yr' 

Therefore,  by  172,  the  n  —  1  functions  Zg?  2:3,  z^  are  connected 
by  a  linear  relation,  i.e., 

0.2Z2-^asZs-\ f-an^n=0.  (7) 

Dividing  (7)  by  2/1^,  and  restoring  the  values  of  Z2,  z^,  •••  2;,„ 

Integrating  (8),  we  find 

«i2/i  +  «2y2  +  «32/3  H h  a„2/n  =  0.  (9) 

Therefore  assuming  that  if  the  Wronskian  of  n  —  1  func- 
tions vanishes,  the  functions  are  connected  by  a  linear  relation, 
we  have  shown  that  when  the  Wronskian  of  n  functions  van- 
ishes, the  functions  are  connected  by  a  linear  relation.  But 
the  assumption  is  obviously  true  for  two  functions,  hence  the 
theorem  is  true  universally. 


APPLICATIONS   AND   SPECIAL  FORMS. 


213 


Linear  Substitution. 
177.    If  the  n  functions  (one  or  more) 

/g  =  O2I  a^2  +  <^22  ^2  H +  a^n^n 


(1) 


are  transformed  into  functions  of  2/i»2/2?  '••  Vn  ^J  the  following 
linear  substitutions,,* 


«1  =   ^l2/l  +  &12  2/2   +    •••    +   Kyn 
^2  =   &212/1+&222/2   H h    &2u2/n 

»n=  &«iyi  +  &n22/2  +    •'•    +   &„„?/« 


(2) 


the  determinant  I  6i„  I  of  the  system  (2)  is  called  the  modulus 
of  transformation.  If  the  modulus  is  unity,  the  substitution 
is  unimodular.  If  a^j,  iCg,  •  •  •  »«  are  independent,  the  modulus 
cannot  vanish. 

178.    If  the  functions  (1)  are  transformed  by  means  of  (2) 
into 

/i  =  mn2/i  +  mi22/2H (-Wi„?/„  1 

/a  =  msi  2/1  +  m22  2/2  H f-  m,,,  y. 


(3) 


frf=  l^nlVl  +  Wln22/2  H h  W„,.2/„ 

the  determinant  of  the  system  (3) , 

Imn  W22  •••  m„^l, 


*  The  learner  can  understand  the  importance  of  linear  substitution  by 
noticing  that  such  a  substitution  is  the  process  involved  in  transformation 
of  coordinates  in  Geometry. 


214 


THEORY   OF  DETERMINANTS. 


equals  the  product  of  the  determinant  of  the  given  sjstem  (1) 
by  the  modulus  of  transformation.     That  is  to  say, 

\m,^\  =  I«,J  X  |6i«I. 

This  is  proved  as  follows.     The  coeflficient  of  y,^, 

Wl«  =  ttii  6u  +  «,2  ^2*  H h  Clin  &«*» 

is  found  by  multiplying  equations  (2)  by  ciii?  ^f2»  •••  «m?  respec- 
tively, and  adding  by  columns.     Whence,  by  53,  we  see  that 


will 

mi2   •••   min 

= 

«n 

ai2 

77121 

m22   '"   rn^n 

«21 

^22 

... 



... 

... 

m,a 

w„2   •••   m„„ 

a. 

a«2 

«2h 


^1       hi     ••'      hn 
621        622      •••      hn 

&„i     b„o    ...    6„„ 


179.    If  /(iCi,  aJa, '"  x^)   is  to  be  so  transformed  by  the  sub- 
stitution 

i»l  =  /8„  2/1  +  A2  ^2  +  —   +  /?l„  Vn    ] 
i«2  =  /?21  2/1  +  /?22  2/2  H h  Au  2/n 


' 

5„=  ;8„i^i  +  ^„22/2  -I (-  ;3„„2/„  J 


(a) 


that 


2// +  2/1  +  -  +  ^n' =  ^1' +  a:/ +  -  +  a^,f, 


the  linear  substitution  is  called  orthogonal.     The  coefficients  of 
an  orthogonal  substitution  must  satisfy  the   following  condi- 
tions. 
A.    Since 

yi'+yi-h'-'+Vn' 

=  (/3ii2/i+ft2  2/2  +  -+A»2/«)'+(Ai2/i+&2/2  4-  -  +ftn2/n)^ 

+  +(/8«l2/l+)8„22/2+-+)8nn2/u)' 

+  •••  +  22/12/2 (ftii8i2  +  ft,&+  -  +i8„i;8„2)  4-  -, 


APPLICATIONS  AND   SPECIAL  FORMS.  215 

we  must  have 

X      fAf    +Af     +'"  +  ^J     =1 

lA^A*  +  AA*-f  •••+A.-/5«.  =  0   (t,  A:=l,2,  ...  n). 

B.  If  we  wish  to  return  to  the  original  function  from  the 
transformed  function,  we  must  put 

For  from  (a)  we  readily  find 

Aia^i  +  /32i^2H hAu^^n 

=  2/i(Ai  Ai  +  Ai  Ai  +  -  Ai  AO  +  2/2(^2  Ai+&/32i+  -  +/5„2  A,) 

+   -    +  ^nCAn  A.  +  An  Ai  +  -  +  An  AO  ' 

Now,  by  I.,  the  coefficient  of  ?/j  =  1,  and  the  other  coefficients 
vanish. 

C.  The    square   of    the    determinant    of    the    S3'stem    (a) 
(modulus  of  transformation)  is  unity. 

For 

Al         A2      -       A: 

Ai    fe  -  A, 


III. 


=  IAJ^=IA«!. 


Al  A2  -   An 
\Din\  is  a  symmetrical  determinant  by  108 ;  since,  by  I., 

A*=o,    A=i, 

the  truth  of  III.  is  obvious. 

D.    Bi„  being  the  minor  of  ySf^  in  I  y8i„  I ,  we  find 

^.■*  =  AJA«1- 

For  multiplying  the  equations 

AiA*  +  -  +AiA*  =  o, 

A*A*  +  -+A*A*=i, 
...        ...        ...       ...J 

A»A*  +  -+AnA*  =  o, 


216  THEORY   OF   DETERMINANTS, 

ill  order  by  Ba,  Bi,,  -•' Bi^,  and  adding,  we  have 

But  all  the  coefficients,  except  the  coefficient  of  ^i^,  vanish ; 
hence 

IV.  5a=AJAnI- 

E.   By  the  preceding  condition  IV., 

(Ai^a  +  -  +  A-u/?*n)  I  AJ  =  Bal3,i  4-  -  +  An/3*«. 

The  second  member  of  this  equation  is  IjSiJ,  or  0,  according 
as  i  and  k  are  equal  or  unequal. 
Whence 

1  Ali^H  +  A2  ^2  +  •  •  •  +  An)S,.  =  0. 

i^.    The  following  relation  holds  between  the  minors  of  the 
modulus  of  the  orthogonal  substitution. 


VI. 


Pr+l  r+1  Pr+l  r+2 
Pr+2  r+1   Pr+2  r+2 


Pn  r+1  Pn  r+2 

For,  by  61, 


lAnI    X 


Al       A2 
Al       ^822 


A 


Ai    y8^  -   Ar 


J521     i522    ...    B,, 


B. 


Brr 


=    lAnl 


r--l 


A+l  r+1       A+1  r+2 
A+2  r+1       Pr+2  r+2 

A  r+1  A  r+2 


An 


APPLICATIONS   AND  SPECIAL  FORMS. 
Now,  by  ly., 


B.21       B.22      '"      B^r 


B^,     Bo 


B,, 


=   I  An  l^  X 


Ai    fe 

P21     P22 

f^rl        A2 


21T 


Whence,  equating  the  second  members  of  these  two  equa- 
tions, the  relation  VI.  follows. 


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